Global Analysis: Differential Forms in Analysis, Geometry, and Physics (Graduate Studies in Mathematics, V. 52)

we see that V corresponds to the vector field the map h, we obtain the formulas (1)

(2)

ar 49

h.

ral ar

hCl ap

a

=

a_ 1,

smV5X2

r

a a -rsincp+ rcoscp ax2 axl

+ 7X1

_

251 57X2

,a -x2a axl +x 57X2

3. Vector Analysis on Manifolds

58

Differentiable functions f : Mk -- R can be differentiated with respect to a Mk vector field V. At a fixed point x E Mk we choose a curve ry : [0, EJ satisfying the initial conditions y(0) = x and y(0) = V(x). The derivative off at the point x in the direction of V(x) is now defined by the formula

V(f)(x) := tf o y(t)It=o The result is a C30-function V(f) defined on the manifold Mk. In the next theorem we stunmarize the properties of this differentiation:

Theorem 7.

(1) (V+W)(f) =V(f)+W(f); (2) V(f1 + f2) = V(f1) + V(f2);

(3) V(f1 - f2) = V(fl) - f2 + f1 - V(f2); (4) If the vector field V is represented as V = local chart, then

V'(y)818yi in some

k

V(f) _

Vi(y)8(foh) i=1

Proof. We will prove (4); all the other claims follow immediately. If the point x E Mk corresponds to the point y E V under the chain map h : V Mk, then -y(t) = h(y + t(V 1(y), ... , Vk(y))) is a curve in Mk satisfying the initial conditions -t(O) = x and ry' (0) = V(x). The formula for V(f)(x) to be proved then follows from the chain rule: k

V(f)(x) = d dt f o9 (y+t(V1(y),...,Vk(y))) _

i=1' V'(y)a(f

h).

0

3.2. Differential Calculus on Manifolds

59

Next we will discuss the notion of Riemannian metric on a submanifold Mk

of euclidean space. The scalar product in R" is denoted by (v, w). We restrict it to the tangent spaces of the submanifold. Definition 7. Let M11k C R" be a submanifold. In each tangent space T, Mk the formula

g= ((x. V), (x. w)) := (v, w)

defines a scalar product. The family {g.,} of all these scalar products is called the Riemannian metric of A1k.

In a chart It : V _ Alk, the Riernannian metric is locally described by the functions g,j defined on the set V, a

a

gij W = A(y) (ay' ayj

)

= (ayi ah,

ah) ami

Thus, the components gij(y) of the Riemannian metric are the scalar products of the partial derivatives of the chart map h considered as being vectorvalued. When it is clear to which chart we refer, h is often omitted (as in the examples to follow). The (k x k) matrix (g,3) is symmetric and positive definite for each y E V. By g(y) we will denote its determinant, which always is a positive function. Recall that we already agreed in Chapter 1 for bilinear forms to denote by (g`J(y)) the inverse of the matrix (g,j(y)).

Example 12. In euclidean coordinates on R", the entries gij(y) = 6,j are constant and equal to one for coinciding indices i = j. and vanish in the other cases.

Example 13. In polar coordinates on R2 - {0}, equations (1) and (2) immediately lead to the following coefficients for the usual euclidean metric:

a s

9r. _ (ar

Or

d a_= 0,

_

_ 1,

,

gw;v

Or 8y7

g,-"

/aa , a

r

2 .

In particular, in polar coordinates on the plane we have g(r, p) = r2.

Example 14. On the sphere S2 - {N. S} C R3 without north and south pole, we consider spherical coordinates defined by

h(y^, y) = (cos f cos ri', sin, cos t,, sine) ,

0 < V < 27r, -ir/2 < z, < r/2.

Computing the tangent vector fields yields a

a

(3)

(4)

a

az-

- Cos

siny

a

axi

+ cos

cos y a-2

- sin yo sing

a 2

,

+ cosy

a 3.

3. Vector Analysis on Manifolds

60

For the coefficients of the Riemannian metric on the sphere we thus obtain

g 4_90b-)=cos29 - /(Y

0, 9vw=

=1.

The determinantt of the metric on the sphere in these coordinates is equal cost 0; therefore, the spherical coordinates degenerate at the to g(v, poles, which is why we deleted them.

We will use the Riemannian metric g of a manifold in order to associate with every smooth function a vector field, the so-called gradient. For a fixed point x E Mk and a tangent vector v E TIMk we first choose a vector field

V such that V(x) = v. The assignment v - V(f)(x) determines a linear functional TxMk - R on the tangent space, and hence there exists a vector grad(f)(x) E TTMk such that the equality

V(f)(x) = gx (grad(f)(x), V(x)) holds for all vector fields. The vector field grad(f) is called the gradient of the smooth function f : AIk -+ JR.

Theorem 8. Let f : AIk -+ JR be a smooth function, and let h : V _ Mk be a chart. Denote the coefficients of the Riemannian metric in this chart by gij. Then k

e(f o h) ij a ayi 9 ayi

grad(f) = i j=1

Proof. Inserting the right-hand side into the definition, we compute k

ij=1

a ayi ,aym

a(f o h) 9ij a ayi

9

o 11)bi _ E a(fayio h) 9ij 9jm - kLa(fayi

m

i,j=1

i=1

o h) = a(faym

According to Theorem 7, the last expression is precisely (8/aym)(f ), which by the definition of the gradient implies the result.

Example 15. The formula for the gradient of a function defined on an open subset U c R" of euclidean space expressed in cartesian coordinates is Of

(5)

grad(f) = n a-i aTi a i=1

Example 16. In polar coordinates on R2 - {0} the coefficients of the metric are

grr = 1, 9rp = 0, gp' =

1

T2

3.2. Differential Calculus on Manifolds

61

For a function f in the variables r, gyp, the formula of the preceding theorem yields (6)

(f) = Orof g ar + of ;, a rr a

of a

1 of a

app = ar ar + r2 aap aV

Example 17. In spherical coordinates on S2 - IN, S} we have g'v;v =

95'"

= 0,

1,

cos y)' 2 and for the gradient of a function f : S2 - {N, S} -R this leads to

Of a

grad(f) = cost ao a0 _ aV For example, the height function f (x1, x2, x3) = x3 on the sphere may be (7)

written in spherical coordinates as f (cp, ') = sin rP, and thus

grad(f) = cos z'

.

Hence the vector field grad(f) on the sphere S2 vanishes at precisely two points, at the north and the south pole.

Vector fields and the Riemannian metric can be represented in various coordinates. Now we derive the transformation formulas for its components with respect to different charts. To this end, fix two charts h : V -> Mk and h : V -+ Mk and denote the points from V by y = (y1, ... , yk) and those from V by z = (z1, . . . , zk). The chart transition maps will be denoted by 0 = (01, ... , 0k) := h-1 o h and t' = (VG1, ... , aiik) := h-1 o h, respectively.

3. Vector Analysis on Manifolds

62

Theorem 9. (1) Let V be a vector field and denote by V 1,

... Vk its components

in the chart h and by 0, ..., Vk the components in the chart h. Then, for each index I between 1 and k. V, (Y)

,

k

(4(y)) az; (b(y)) i=1

(2) Denote by g;j the coefficients of the Riemannian metric with respect

to the chart h and by g,; those referring to the chart h. Then they are related by the formula 01 corn

k

gi;(y) _

ay;

91m (0(y))

-,

l,m= I

In particular, for the determinant of the Riemannian metric the following holds: 2.

9(y) = det

Nr

9 (4(y))

1

Proof. The first formula is a consequence of the chain rule. In fact, = a k atvr a _ k awl a ( h of h` az' az' az' az' ax' ayl ) = h.

(a) =

E

(E

a)

a-

This leads to the following formula, describing the vector field in the chart h:

,a

k

a2

i=1

k 1=1

k

\ i-1

a

ax / a

The second formula follows directly from the first: a

a\

ax'' axj

2 k

1,m=1

a a az' azi C ayl ' aym > =

811,1 awm

E A

1,m=1

atll aij,m

az' azi

if, in addition, the (formally completely equivalent) roles of h and h are exchanged.

We will use these transformation formulas to define the divergence of a vector field. This will be taken to be a function on the manifold determined by a local formula. Therefore, it has to be proved that the corresponding expression is independent of the choice of the chart. Definition S. Let V = F; V'(y)a/ay' be a vector field on the manifold Mk expressed with respect to the coordinates of the chart h : V Alk, and let

3.2. Differential Calculus on Manifolds

63

R be the determinant of the Riemannian metric in this chart. g(y) : V Define a function div(V) : h(V) R. the divergence of V, by the formula

1

div(V)(h(y))

k

a (v/g V M) ayi

Vg i=1

Theorem 10. The defining formulas for the divergence of a vector field for

Mk and h : V -- Mk of the manifold Mk

two arbitrary charts h : V

coincide at corresponding points: a(,/g-V'(y))

=

1

k i=1

1

- k a(vg= f7 '(z))

azi

8yi

The proof uses a formula expressing the derivative of a determinant by its minors. For the sake of completeness we state it here.

Lemma 1. Let k2 functions hj(x) be defined on an open subset V of lRc. Denote by H(x) := det[hij] the determinant and by A;j(x) the minors of the matrix hij(x). Then k

_H(x) _ ax''

Ohij

axr

Di (x) . J

Proof. Differentiating the determinant and applying the Leibnitz rule yields e

OH(x)

ax''

h12

...

hlk

h11

...

hl.k-1

a=te

hkl

...

hk,k-I

a8x'

+...+det

det 8hk1

hk2

...

hkk

Expanding each of the k resulting determinants with respect to the column 0 containing the derivatives leads to the stated formula.

Proof of Theorem 10. Applying the preceding formula, we obtain 1

9y)

k

a( 9(y) V'(y)) = ayi

;=1

k

aV'(y)

;=1

ay

+

k

i=1

V i(y) ai aa/

i

= i=1 L yi

My)) j=1 m

V'(y)ay; In 9(O(y)) +In Idet gyr

+ i=1

(in

g(y))

3. Vector Analysis on Manifolds

64

Now we compute the partial derivative of the first sum and simplify the resulting expression, making use of the following formulas: k

aoi

at' = alJ . s i azj

i=1

This immediately leads to the expression k-

1

9 5) ;=1 r--

k-

a_ 9(1/) V`(}/)) =

syi

9(z) O(z))

O(

1

azi

Vr9(z)

k

2

+ i,j,t=1

i

m

1

'(4(y)) a )az,

i

+ azi

In

1

Idet ayr I 0

Last, we have to show the following equation for each index 1 < j < k: k i,1=1

8

m

l

ai

z1

Oyi

az,j In Idet 00

+

T

= 0.

Io

Using the lemma, the second summand in this equation can be written as

a

jln

-1 k

det amm I o V, = (det I

a

o 10

V,) -1

= (det syr J o L

k

i,a.0=1

J

L

(det 00m) OI`

J

J

a4,i

a2Q°

0° aZ;

syiaya

The matrix [Ow'/ad] is the inverse of [490m/ay''], and, by Cramer's rule. a-j0 m 1 0 wl -1 = D&Q . (det I 8z° This yields aim OP a20° a rk In Idet az"

ay*

-

i,°,)9=1

I

and the equation to be proved becomes k 0n2jpi 00° kk

i,° =1

ayiay,3 azi 19z°

0-20a

atpi 0,03

ayiay'3 azi az° = 0

azjaz° ayi +

But this identity immediately follows from the fact that p and inverse maps: Differentiate the equation k

n

i

ww) i=1

.

a;

are mutually

3.2. Differential Calculus on Manifolds

65

with respect to the variables z1, set a = 1, and finally take the sum.

0

Example 18. In cartesian coordinates on R", the formula for the divergence becomes "

aVi axi

div(V) _

(8)

Example 19. In polar coordinates on 1R2 - {0}, since g = r, for each vector field V = V' 010r + V2a/0' the following formula holds: (9)

div(V) =

1

a(rV')

r

+

a(rV2) a

=

l

OV1

Or

1

8V2

r

aV

+ -V1 +

Example 20. In spherical coordinates on S2, the determinant fo the metric is g = cost ly and thus for every vector field V = V1a/a

div(V) =

1

cos

y

(

a(cosip V1)

0p

a(cosii - V2)

+

aV1

av2

) - acp + a

a

2

- tan tp V .

Theorem 11. Let V be a vector field and f a smooth function on Mk. Then

div(f V) = f div(V) + V(f). Proof. It suffices to prove the formula in an arbitrary chart. This is done by a straightforward computation:

div(f V)

1

" i=1

8(,g- f Vi) _ agi

1

"" i=1

of '

09

V+ f

= f div(V)+EViaf = f div(V)+V(f).

a(Vfg- y')

ay O

i=1

The last operator to be introduced in this chapter is the Laplacian 0; it acts on functions and is a second order differential operator.

Definition 9. Let f : Mk --' R be a smooth function. Then the Laplacian of f is defined as a function on Mk by the formula

O(f) := div(grad(f )) Remark. The operator A : C°O(Mk) C- (Mk) is a linear operator acting in the vector space of all C0°-functions on Mk. In mathematics, there are-concerning the sign of 0-two differing conventions. Classically and

mainly in analysis, A is defined as above. On the other hand, in various branches of geometry and global analysis it is common to introduce the Laplacian via the formula A(f) = -div(grad(f)) and to call this the Laplace-Beltrami operator. There are good reasons to do so, as we will see

3. Vector Analysis on Manifolds

66

in connection with the classical integral formulas. Thus, studying a text, the reader has to be careful to detect which sign convention was chosen by the author. In this book we decided to use the particular choice of sign for the Laplacian stated in the definition.

Theorem 12. The Laplacian 0 : C-(A1k) -. C-(Mk) is a linear operator. Moreover, for two functions fl and f2 the following formula holds:

A(fi - f2) = fi ' (f2) + f2 A(fl) + 2 (grad(fl ), grad(f2)) Proof. The linearity of 0 is obvious, and the second formula follows by combining the corresponding rules for the operators grad and div:

A(fi f2) = div(grad(fi f2)) = div(fi grad(f2) + f2 grad(f1))

= fi div(grad(f2)) + f2 div(grad(f1)) + 2 (gr'ad(fi). grad(f2))

0 In local coordinates we immediately obtain from grad(f)=Ya(foh)i;o

the formula

AM _ I-

s

a k

i a(foh)

g

Again, we write out the explicit form of the Laplacian in the examples we have discussed so far.

Example 21. In cartesian coordinates on R", (11) .=1

(OXi)2

Example 22. In polar coordinates on Ilt2,

(12) o(f) =

8 (Of

a(r

r ar lrar) + app \

Example 23. In spherical coordinates on S2, (13)

f

cost 0 OW2 + f,2

1 of 102f are + r ar + r2

a2f

Of

r2 app))

-tan

f

In these coordinates the height function f (x1, x2, x3) = x3 on S2 can be represented as f = sin ip. This leads to

A(f) = -2sing(, = -2f,

3.3. Differential Fornu; on Manifolds

67

i. e., f is an eigenfunction of the Laplace-Beltrami operator corresponding to the eigenvalue -2. Other eigenfunctions of A on S2 are constructed starting from a harmonic and homogeneous polynomial P(xl, x2. x3) in 1R3, 0R' P = 0, and restricting this to the sphere S2. The resulting function is an eigenfunction of the Laplacian (see Exercise 25).

3.3. Differential Forms on Manifolds Up to now, for every manifold Al" we considered its tangent space TAI' and vector fields. Now we go one step further and form the exterior power AA(T.Mm)

of all k-forms w= : TTM' x ... x T1Mm - IIt at the point x E Alm.

Definition 10. A k-form wk on a manifold /L1m is a family {wi} distinguishing a k-form wT E Ak(Mt) at each point x E AI'".

The differential of a smooth map f : N" -i M"' between two manifolds allows to pull back k-forms from Mm to yield k-forms on N". This is accomplished via the formula

(f'wk)(vi, ... , vk) = wk(f.(vl), ... , where vl, .... vk E TyN" are tangent vectors to N' at the point y E N", and f.: TyN" Tf(y)Mm is the differential of f at this point. The k-form f `wk is called the induced form or pullback of wk by the map f on N". This construction can, in particular, be applied to a chart h : V Alm of the manifold Alm. Hence, for a fixed chart., to every k-form wk on AI'"

there corresponds a k-form h' (wk) on the open set V in the space R' or Mm, respectively. If y = (yi, ... , y'") are the associated coordinates, then h'(wk) can be represented by means of the component functions w/ as

h*(wk) = Ewldy1. I The summation extends over all ordered multi-indices I = (il ... << ik), where dy1 is a shorthand notation for the k-form dy1 := dy`' A ... A dy'k.

Definition 11. A k-form wk on the manifold Mm is called a differential k-form or smooth k-form if for each chart h : V - Mm the coefficients w1 of the k-form h*(wk) are smooth functions on the subset V C IIt". Differential forms can be added and multiplied by functions defined on A1m in an obvious way. Hence the set S2k(Alm) of all C°°-forms of degree k is a module over the ring C°O(AIm) of smooth functions on the manifold.

3. Vector Analysis on Manifolds

68

The exterior derivative of k-forms-familiar from the euclidean space ;,F" and discussed in Chapter 2-can now be transferred to the situation of kforms on manifolds without difficulties, preserving all the known properties.

This proceeds as follows: For a k-form wk on Al' and a chart h : V - 11"' we first consider the induced form h'(wk) and its differential d(h'(wk)). The latter is a (k+ 1)-form on the set V, and its pullback under the inverse chart map h-1 : It (17) -- V is a (k + 1)-form. This yields a (k + 1)-form

dwk := (h-1)`(d(h*w')) defined on the open set h(V) C 1M'". The construction just described is independent of the particular choice of the chart, and hence it uniquely defines

a global (k + 1)-form dwk on the manifold M. In fact, for another chart It, V1 -. Mm satisfying h(V) fl h1(V1) 0 0, we obtain on the intersection :

h(V) n h1(V1) the equality

(hl 1)'(d(hiwk)) =

(h-')*(hi 1 o

h)'(d((h-1 o hl)`h'(wk)))

= (h-1)d((h, 1 o h)'(h-1 o h1)*h'(wA)) = (h-1)d(h*wk) . This computation relies on the fact that the exterior derivative commutes with the diffeomorphism (hi 1 o h), a property discussed in Chapter 2. Definition 12. The (k + 1)-form dwk defined starting from the k-form wk on Al... is called the exterior derivative of wk. All properties of the exterior derivative known from euclidean space remain valid in the situation of a manifold. The next theorem summarizes them.

Theorem 13. For arbitrary forms wk,

Wk' 17

I

on a manifold M.. the follow-

ing properties hold: ( 1 ) d (wk + wk) = dwk + dwi ;

(2) ddwk = 0;

(3) d(,,;k A tl') _ (&,k) A,/ + (-1)Awk A (d77');

(4) if f : N" -+ Mm is a smooth map, then f' commutes with the exterior derivative, d(f*wk) = f'(dwk) The purely algebraic operation of forming the inner product between a vector field V and a k-form wk is also transferred into the situation that both objects are defined over a manifold.

Definition 13. We define the inner product of a vector field V and a k-form wk by

(V _J wk)(W1, ...,Wk-1) := wk(V,W1.....Wk-1).

3.4. Orientable Manifolds

69

3.4. Orientable Manifolds An orientation of a real vector space is the choice of one of the two equivalence classes in the set of its bases discussed in the first chapter. This can, in particular, be applied to the tangent space T=M' of a manifold, and leads to the notion of an orientation 0= at the point x E Alm. An orientation of a manifold Mm consists in a "continuous" choice of orientations at each of its points.

Definition 14. An orientation 0 of a manifold Alm is a family 0 = {Or} of orientations in all tangent spaces TM m depending continuously on the point x in the following sense: At each point x E Alm there exists a chart h : V Mm containing this point such that the basis {h. (a/ay' ), ... , h. (a/aym) } is compatible with the orientation Oh(y) for every point y E V.

Definition 15. A manifold Al' is called orientable if there exists at. least one orientation on it. First we state a necessary and at the same time also sufficient condition for the orientability of a manifold in terms of the chart transition maps.

Theorem 14. Let 0 be an orientation on Mm. Then there exists a family {(hi, V )liEI of charts with the following properties: (1) The image sets hi (Vi) cover the manifold,

film = U hi(V) ; iEI

(2) If the intersection hi (V) n h; (V3) 34 0 is non-empty, then the determinant of the differential of the chart transition map h, 1 o hi is positive,

det[D(hj 1 o hi)] > 0. Conversely, if there exists a family of charts with these properties, then Mm is orientable.

Proof. Let 0 be an orientation on Mm. Choose those of the charts h : V Alm for which the basis {h.(a/ayl ), ... , h .(a/ay`)} is compatible with the orientation of the manifold. By the definition of an orientation these charts cover the manifold. For two of these charts, hi : V

Afm and

h3 : Vj -+ Mm, with coordinates y = (y1, ...Iym) and z = (z'...., zm) we have at mutually corresponding points

{hi.(a/ayl), ... , hi.(a/ay')} = Oh;(y) = Oh,(z) _ {hj.(a/azl),...,hl.(a/azm)}.

3. Vector Analysis oil Manifolds

70

Therefore, these bases are compatibly oriented, the transition matrix between them is the differential D(h, o h;), and we obtain 1

det[D(hj 1 o h;)] > 0. The converse is proved analogously.

If the manifold All C IR" is described by (n - m) independent equations, then it is orientable. We will prove this fact now.

Theorem 15. Let f1, .... f"_,,, : U , R be smooth functions defined on an open subset U C 1R". and assume that

... A dfn-in 1 T0 at each point. Then the manifold df1 A

Mm = {rEU: fl(x)_...=fn-n(x)=0l is orientable.

Proof. Consider the euclidean gradients grad(fl ), ... , grad(fn_,,,) of the functions. By assumption, these vector fields are linearly independent at each point of the set U and, moreover, perpendicular to the tangent space Tjlll"' at the points of the manifold (compare Theorem 5). An orientation Ox in TiAfm is distinguished by requiring that a basis v1, ... , vm E TAf"' is positively oriented if and only if

d]R"(grad(fl)(x) ....grad(fn-,.)(r),v1, ...,v, ) > 0. Here dR = dx1 A ... A dx" denotes the volume form on R". It is not hard to see that this condition determines all orientation on M"'. An oriented submanifold Al'" c R" carries a distinguished differential form of highest degree, the so-called volume form. Choose a basis e1.... , em E TxM"' consisting of mutually perpendicular vectors of length one in the fixed orientation Ox at any tangent space TIM . For arbitrary vectors ill i .. , v,,, E Tx:'ll 'n we define

(vi,el),

dill'"(v1....,vm) = det (v,, el), This definition uniquely determines the form dAl". Every other basis in the same orientation can be represented as a linear combiel, nation e; = F_ A,2ej with an orthogonal matrix A of positive determinant. But then det(A) = 1.

Definition 16. The m-form dMm on the oriented manifold Mm C R" is called the volume form.

3.4. Orientable Manifolds

71

Remark. The volume form dAlm is not the exterior derivative of an (m-1)form. Nevertheless, this form is traditionally denoted by dMm. The volume form does not vanish at any point of the manifold. Evaluating dMm on any orthogonal basis e1, ... , em in the orientation yields dMm(e1, ..,em) = 1.

On the other hand, the orientation can be reconstructed from the volume form. In fact, a basis v1, ... , vm is positively oriented if dt11m(vl, ...,vm) > 0. Changing the orientation of the manifold results in a change of sign for the volume form.

Theorem 16. An m-dimensional manifold Mm is orientable if it carries a nowhere vanishing differential form of degree in.

Proof. For an orientable manifold Alm the volume form dAlm has the necessary property. Conversely, suppose that there is an m-form Wm on AI"' not vanishing at any point. Then we call a basis v1, ... , vm positively oriented if

Wm(vl, ...,v",) > 0. This determines an orientation on Al m.

0

In a chart h the induced volume form h*(dM'") is proportional to dy' A ... A dy"', h' (dMm) = f (y)dy1 A ... A dym. We compute the function f (y) as follows:

f2(y) = (h`(dMm)(a/ayl, ... , a/aym))2 = det2 [(h.(a/ay'), e,)] = det [(h.(a/ay'),ej)J det [(ej,h.(a/ay'))] = det [(h.(a/ay'), h.(.9/ay'))] = det[g;l] Hence f2 (y) is equal to the determinant det[g;j (y)] of the Riemannian metric, and we obtain the formula

h*(dM-) =

9(y)dy' A...Adym. Example 24. For a surface piece A12 C R" with a parametrization h ][1;2 , AI2 C R", the following classical notation is frequently used:

/ ah ah \ E = 911(x,y) = ate, YG = 922(X, Y)

F = 912(X, y) =

ah ah

(

-.9Y

09-Y

ah ah

a-, 5

3. Vector Analysis on Manifolds

72

The formula for the volume form is then

EG-F'2dxAdy.

dM2 =

Example 25. The volume form of 1R' in cartesian coordinates is

d1W = dx1 A... Adx".

(14)

Example 26. In polar coordinates on 1R2 - {0} we have g(r, gyp) = r2, and hence the volume form is

dIR2 = r . dr n dip .

(15)

Example 27. In spherical coordinates on the 2-sphere, and thus the volume form is

1P) = cost v/i,

dS2 = cos t dip A diP.

(16)

In the case of an oriented manifold, the divergence of a vector field V can be expressed in terms of the exterior derivative and the volume form. Theorem 17. Let Mm C R" be an oriented manifold, let dMm be its volume form, and let V be a vector field with divergence div(V). Then

d(V J dMm) = div(V) dM'". Proof. In local coordinates we start from m

Vs(y)a

V=

y

i=1

and

ii

dMm = /dyl

A...Adyt.

This implies m

V-i dMm =

,Fg

(-1)'-1V'dy' n...AdyiA...dy'", i=1

and the formula

d(V.j dMm) =

m

a(/ V t) dy' A ... A dym = div(V) dMm

i=1

immediately follows from the definition of the divergence of a vector field.

0

Remark. As in the euclidean space R. for every (oriented) manifold M'" the 1-form w , dual to a vector field V can be described using the volume form and the Hodge operator:

*wv := V i dM' . The divergence formula can then be written as d(*wl,) = div(V) dM'".

3.4. Orientable Manifolds

73

Non-orientable manifolds exist; proving non-orientability for a particular manifold, however, is sometimes a little more difficult than to show orientability. We state a simple criterion for the non-orientability of a manifold. This can be used, e.g., to prove that the Nlobius strip is non-orientable.

Theorem 18. Suppose that. for a manifold Mm, there exist two charts Mm and h : V

h:V

M' with the following properties:

(1) V and f7 are connected sets, and the intersection h(V) n h(V) is not connected;

(2) the determinant det(D(h-' o h)) has opposite sign at two points in

h-'(h(V) n h(V)). Then M' is a non-orientable manifold. V

Proof. Assume that MI is orientable. The sets V and V are open subsets of ]R'", and hence orientable. Thus, without loss of generality, we may assume that the chart transition maps h and h preserve the orientations. Consider the volume form dMt, and represent it in both coordinate systems:

h* (dM-) = J dy' A ... A dyt , h' (dM-) = f dz' A ... A dz' . Note that in both these cases the sign is "+", since the chart transition maps h, h preserve the orientation. Because

(h-1 o h)'(dz' A ... Adz') = det(D(h-' o h)) dz' A ... A dzm, we have

f det(D(h-'oh)) = f. But this is a contradiction, since f and Vg' are always positive.

0

The orientation of a manifold induces a unique orientation on its boundary. To define it, we make use of the exterior normal vector field. Its definition relies on the following observation. Lemma 2. Let U C ]H' be an open subset of the half-space. let x E U n 8HHm be a point in the boundary OHHt, and let f : U - V be a die`eomorphism from

U to an open subset V C H. F o r a vector v = (x, (v', ... , v' )) E T=H"'

3. Vector Analysis on Manifolds

74

at the point x with non-positive m-th component, v"' _< 0, the image vector MV) = (f (x), (w1, ..., w"')) also has non-positive m-th component w"' < 0.

f. (v)

Proof. Choose a straight line -y(t) = x + t v and note that, because of v"' < 0. the point y(t) belongs to U fl HH"' for sufficiently small negative values of the parameter t. Then we have y(0) = x, y(0) = v. and the in-th component of f.(v) becomes negative:

_dtf"'(x+tv) d

< 0. = lim f"'(x+tv)-0 t t-o-

0

At a boundary point x E 8M' C M' of a manifold there are two tangent spaces. On the one hand, there is the tangent space TxMm of Al'. and, on the other hand, the boundary determines its own (in - 1)-dimensional tangent space, T=(8M'") C TXM'". At each boundary point x E 8M'. we define a unique tangent vector N(x) E TxMm by the following conditions:

(1) N(x) is perpendicular to T (2) N(x) has length 1; (3) for a chart h : V C Him - Mm around the point x E Al"' the m-th component of the vector (h)-1(N(x)) is negative. Definition 17. The vector field N constructed along the boundary is called the (exterior) unit normal vector field of the boundary.

3.4. Orientable Manifolds

75

We fix an orientation of the boundary OMm of an oriented manifold M' by calling a basis vl, .... vm_1 E TT(OMm) positively oriented if the extended is positively oriented in T2A1m. A simple argument basis A((x), vi, shows that this condition actually determines an orientation of the boundary. For the corresponding volume forms we have the important formula:

Theorem 19. The volume form of the boundary of a manifold is the inner product of the exterior normal vector field N with dull'":

d(aMm) = Ni dM'n Proof. Choose an orthonormal basis in the orientation of the vector space T,,111 consisting of the normal vector el := N(x) together with additional vectors e2, ... , e,n. For arbitrary tangent vectors v1.... , 0m-I to the boundary we obtain

(Ari ditlm)(t'1, ...,v1) = d111"'(el,vl, ...,vrn-1) (vi, e2)

...

(vl,em)

(vm-1, e2)

...

(vm-1, em)

= det = d(OMm)(i'1, .. , l'm-1) , since the vectors vl, ...,v,,,_1 are perpendicular to el.

0

In the subsequent sections of this book the boundary of a manifold will always be oriented this way.

Example 28. The exterior unit normal vector field of the sphere Sn-I(R) of radius R, considered as the boundary of the ball Dn(R), has the form

1r(la +...+x nal ax-1 ax's

N(x)

= R(x

The volume form of Sn-I(R) can also be computed by means of dSn-1(R) _ N.J dR', yielding the formula (17)

dSi-l(R) =

1

E(-1)'-Ix'dxlA...A dx'A ...Adx".

i=1

Hence the volume form of the sphere S"-'(R) can be represented via the embedding i : Sr-1(R) -+ IR" as the form induced from the following (n-1)form defined on Rn:

,,n-1 =

-1x'dxl A ... Adx' A ... Adxn' i=1

i.e., the following equation holds:

dS"-1(R) = i.(wn-l)

3. Vector Analysis on Manifolds

76

Computing the exterior derivative of wi-1 in R' leads to the formula

dwn-1 = n dlRn ,

R and by applying Stokes' theorem from Chapter 2 we obtain the integral of the volume form:

J n-l(R) dSn-1(R) =

JD"(R)

dwn-1 =

Rvol(D°(R)) .

3.5. Integration of Differential Forms over Manifolds The integral of an m-form over an m-dimensional manifold will be defined by dividing the manifold into small subsets which are diffeomorphic to open subsets of R' or Hn', respectively, and integrating the given form using its chart representatives one by one. The sum of the resulting values is then the total integral of the form over the manifold. From the very beginning we will confine ourselves to compact manifolds in order to have to deal with finitely many summands only and thus to avoid convergence questions for series arising otherwise. A detailed exposition of this definition of the integral requires a so-called partition of unity in order not to count contributions from overlapping charts twice. This is a special family of smooth functions to be discussed first. Recall that the support of a function cp : Mn' - R is the closure of the set {x E Mm : V(x) 710).

Theorem 20. Let Mm be a compact manifold. Then there exist smooth functions and charts and hi : V - Mm (1 < i < l)

cpi : A f '- 1R with the following properties:

(1) The support of the function vi is contained in hi (V ), supp(wi) C hi(V) (2) The functions Wi are non-negative, and their sum is equal to one: Ws(x)

1.

i=1

Proof. We choose a chart h., : Vr - M"' around each point x E M n satisfying h,,(0) = x. Here, the Vi are open subsets of Rn or H'n. Choose, moreover, non-negative functions 1G:: R'n, H'n R such that (1)

z(0) = 1;

(2) supp(r!.?) C V.

3.5. Integration of Differential Forms over Manifolds

77

Setting

;Px(y) =

0

if y V hx(VV)

1Gx(hx 1(y))

if y E hx(VV),

,

these functions can be transferred to form a family of non-negative smooth

functions bx : M' --+ R on M. By construction we have c3x(x) = 1 and supp(,px) C hx(VV). Taking advantage of the compactness of Mm, we finally

obtain finitely many functions 31, ... , yet and charts (hl, V1). ... , (h1, VI) satisfying

Ar = I

t

{xEMm:yi(x)j4 01

ii=11 < and supp(cpi) C hi(Vi). The sum i(, is positive at each point, and the functions cpi we are looking for result from normalizing these functions 'A' Vi = cpi/v.

Now we will define the integral of an m-form w' over an oriented and compact manifold M. To this end, we choose charts hl, .. . , ht and a partition of unity, Cpl, .....pt subordinate to them satisfying the properties formulated in the preceding theorem. Furthermore, we suppose that the chart maps hi : Vi -+ M'" preserve orientation. Then

fi(y)'dylA...Adym is an rn-form on V with compact support, and we eventually define the integral: t

wm

J

jhWm)

t=1

t

_

Jf(y).dy1...dym. i

t=1

We show that this definition of the integral is independent of the chosen partition of unity. Consider another atlas (h1, V1), ..., (h,., V,.) with subordinate partition of unity 'p1, ... , ;p,.. If the intersection of the chart ranges is not empty, hi(V) f1 ha(V0) 0, then

fi(0pWm)

=

jhQ.Jm).

This follows immediately from Theorem 7 in Chapter 2, since the determinant of the differential D(h= 1 o ha) is positive. Summing now over the

index i, 1 < i < 1, as well as the index a, 1 < a < r, we obtain from E Bpi =_ E cpa = 1 the relation i

a

a=1

V

f hi (vi ' gym)

gym) _ i=1

V,

3. Vector Analysis on Manifolds

78

Definition 18. The m-dimensional volume of a compact and oriented submanifold Mm C lR" is the number

f

vol(Mm) :=

J M^'

dM'".

Consider a curve, viewed as a 1-dimensional submanifold M1 C IR" of R". If It : (a. b) - M1 is a parametrization of the curve, then

jdM'

b

=

Za

and we recover the length of the curve. More generally, if It : V - If' is a parametrization of Mm, then, using the coefficients of the Riemannian metric, the volume of M' can be written as

vol(M'n) =

det[g,.1 (y)] dy.

JV

Example 29. The last formula in §3.4 shows that the (n - 1)-dimensional volume of the sphere Sn-'(R) and the n-dimensional volume of the ball D"(R), both of radius R, are related by

vol(S"-1(R)) = Rvol(D"(R)). Example 30. The coefficients of the Riemannian metric on the torus of revolution discussed in Example 4 are

(ri + r2 cos V)2,

9vv = 0, g = r2

.

Hence f = r2(rl + r2 cos V), and we obtain 21r 2ff

fJr2(ri + r2 cos cp)dcpdi' = 47,2r1 r2 .

vol(T2) = 0

0

We will conclude this section by a remark concerning measure theory on manifolds. Let CO(Mm) be the ring of all continuous functions defined on the compact and oriented manifold Mm. Setting

µ(f) :=

JMm

f dM-

defines a linear functional µ : CO(Mm) - lR which is positive and monotonously continuous in the sense of the following properties:

(1) If f E C0(lbfm) is a non-negative function, then µ(f) > 0. (2) If f,, is a monotonously increasing sequence converging to a function f E CO(Mm), then lim

n -.a o µ(f")

= µ(f)

3.6. Stokes' Theorem for Manifolds

79

This turns the pair (CO(Mm), p) into a so-called Daniell-Stone functional on the set M. Within the framework of general measure theory, one first associates with every, D-S functional p on a set X an outer measure p' on X. The Caratheodory construction then leads to a a-algebra of subsets in X on which p` becomes a measure. By applying these general principles to a compact and oriented manifold, one constructs the so-called Lebesgue measure on MI. It is defined on a a-algebra containing the Borel sets, and every bounded and Borel-measurable function turns out to be integrable. For the purposes of vector analysis we do not need this extension of the notions of measure and integral to manifolds, since the functions occurring are, as a rule, at least continuous. It is, however, interesting to note that the notion of integral treated here fits into the general theory of measure and integration as sketched above. The interested reader may refer to the literature for details (see the book of K. Maurin in the bibliography).

3.6. Stokes' Theorem for Manifolds In the preceding sections we collected all the notions necessary for the formulation of Stokes' theorem on manifolds.

Theorem 21 (Stokes' Theorem). Let Mk be a compact, oriented manifold, and suppose that the boundary OMk is endowed with the induced orientation. Then, for every (k - 1) -form wk-1 on Mk,

f8Mk wk-1

= f

dwk-1

JAlk

If, in particular, Mk has no boundary, then for every (k - 1) form dwk-1

wk-1

= 0.

AIk

Proof. In Theorem 20, it was shown that we can choose finitely many charts (V1, h1), ... , (Vr, hr) covering the manifold 11Ik, together with a subordinate partition of unity O 1, ... , cpr : Mk R. We label these charts in such a way that, on the one hand, for each index i less than a certain index ro the

set V intersects the boundary, V n 8Mk 34 0, and, on the other hand, for all indices i > ro the set V is disjoint from the boundary, V n OMk = 0. From >i vi = 1 we obtain >i dcpi = 0 and use this to rewrite the exterior derivative of k-1

=

wk-1:

k-1 i=1

-

Pidw k-1 i=1

+

dipi n i=1

wk-1

i=1

80

3. Vector Analysis on Manifolds

This implies the equation r

JM' dwk-1 =

d (hi (Piwk-1))

V is an open subset of Rk for rp + 1 < i < r, and h; is a k-form on V1 whose support is completely contained in V. For each of these indices i we choose a k-chain c in Rk for which supp h, (Wiwk-1) C Int ck C ck C V j.

Applying now Stokes' theorem for chains (Theorem 8, Chapter 2) to ck, we obtain

Jd(h(iwk_1))

wA-1))

=

f

h' (Viwk-1) = 0,

d (ht = Jask since the form vanishes on the boundary of the chain. Now we consider the indices i between 1 and ro. For any of these, V is an open subset of the halfspace Hk, and as before we obtain chains ck in lHlk with the same properties of the supports with respect to h; Applying Stokes' theorem to these as well, we obtain 1.

1v;d (hi Now hi

(,p1wk-1) does

hi (ca1wk-1)

J

.

not necessarily vanish any more on Ock n

(Rk- l x {0} ),

but only at the points of 8c; belonging to the interior of Hk: d (h;

(Vjwk-1))

_

I

VflRk-1

V;

hi

(`Piwk-l).

The pairs (V n Rk-1, h1IRk) with i = 1, ..., ro form a covering of the boundary BAIk. Hence, by the definition of the integral, ro

wk-1

-

t-1 knRlshowing the equation we set out to prove. Jamk

h i (ViwR-1)

,

1

O

In the sections to follow we will be dealing with various applications of Stokes' theorem. As a generalization of the discussion in §2.5, we will first study line integrals and prove an analogue-only for 1-forms, however-of Poincare's lemma. This holds for manifolds in which every closed path can be contracted to a point. Definition 19. A connected manifold Mk is called simply connected if every two C'-curves co, cl : [0, 1] , Mk with coinciding initial and end points are homotopic.

3.7. The Hedgehog Theorem (Hairy Sphere Theorem)

81

By Theorem 9 in §2.5, whose proof immediately carries over to the case of a manifold, on a simply connected manifold the line integral of a closed 1-form w1 depends exclusively on the end points of the curve. Having fixed a point xo E Mk, the line integral along a curve joining the points xo and x, w lox uniquely defines a function on the manifold. Its differential df coincides with w1, and we obtain f(x) =

Theorem 22. Every closed 1-form on a simply connected manifold is exact.

Example 31. The winding form defined on 1R2 - {0} is closed, but not exact. This shows that R2 - {0} is not simply connected.

3.7. The Hedgehog Theorem (Hairy Sphere Theorem) Consider two oriented compact manifolds Mk and Nk without boundary and of equal dimension. Two maps fo, fl : Mk - Nk between them are called homotopic if there exists a smooth map

F : Mk x [0,1] -+ Nk such that F(x, 0) = fo(x) and F(x,1) = fl (x). We prove

Theorem 23. Let wk be a k-form on Nk and let fo, fl : Mk

Nk be

homotopic maps. Then

JMk0 (wk) = fm k fl (wk) . Proof. The oriented manifold Mk x [0, 1] has boundary

8(Mk x [0,1]) = Mk x {1} - Mk x (0}, where the minus sign indicates that the orientation is reversed. Therefore, Stokes' theorem implies

Jf

k

fl (w) -

JMk

f(wk) =

F*(wk) = 8J(Mkx[O,1])

JMk x [OI]

dF(wk) .

But the form dF*(wk) = F'(dwk) = 0 vanishes, since the k-dimensional manifold Nk carries no non-trivial (k + 1)-forms.

0

Theorem 24. The antipodal map from the sphere to itself, A : Sn - Sn, A(x) = -x, is homotopic to the identity Ids. only for odd dimensions n. Proof. Consider on Rn+1 the form n+1

wn =

_

-1x' dx1 A ... A dx' A ... A dx"+1 i=1

3. Vector Analysis on Manifolds

82

whose restriction to the sphere Sn is its volume form dSn (Example 28, equation (17)). If A is homotopic to the identity Ids.., then the previous theorem implies

Jis.

A*(wn) = / n w" = vol(Sn) . S

The induced form A* (,n) = (-1)n+lwn is proportional to the form

n.

Thus we obtain the condition (_1)n+lvol(Sn) = vol(S"), i. e., (n + 1) has to be an odd number.

Theorem 25 (Hedgehog Theorem). A sphere S2k of even dimension has no nowhere vanishing, continuous tangent vector field.

Proof. Suppose that there exists such a vector field on the n-dimensional sphere S". We first approximate this vector field by a smooth vector field V (Stone-Weierstrass theorem), and then normalize it so that the vector V(x) has length one at each point. Next we consider the resulting smooth tangent vector field as a vector-valued function V : Sn - Rn+I satisfying the following two conditions:

(x, V(x)) = 0,

IIV(x)II = 1.

Define the homotopy F : Sn x [0, 1] - Sn from the sphere to itself by the formula

F(x, t) = cos(irt) x + sin(irt) . V(x). The length of F(x, t) is equal to one everywhere, since x and V(x) are perpendicular. Moreover, F(x, 0) = x and F(x, 1) = -x, i. e., F is a homotopy between the identity and the antipodal map of the sphere Sn. But then the dimension n has to be an odd number. In the German mathematical literature, this result is known as the "Hedgehog Theorem" (,,Satz vom Igel"), since its contents can be expressed figuratively by saying that a hedgehog cannot be combed in a continuous way. Because it is so vivid, we prefer this to the name "Hairy Sphere Theorem", which seems to be more common in the Anglo-Saxon world.

3.8. The Classical Integral Formulas Now we will discuss the classical integral formulas already treated in §2.6 for chains. Compared to the preceding case, in this new formulation we benefit

from having notions like divergence, gradient and Laplacian as developed in the differential calculus on manifolds at our disposal. At the same time, the orientation of the manifold and the exterior unit normal vector field on the boundary will play a special role. We will prove these integral formulas

3.8. The Classical Integral Formulas

83

for arbitrary compact and oriented manifolds (in R'). This will be the final formulation of the classical integral formulas as they are needed in many branches of mathematics as well as theoretical electrodynamics and hydrodynamics. We start with the Ostrogradski formula relating the divergence of a vector field to its flow across the surface.

Theorem 26 (Ostrogradski Formula). Let Mk be an oriented, compact manifold, and let N be the exterior unit normal vector field to its boundary. Then, for every vector field V : Mk Tlllk on Mk,

div(V)dMk =

(V, N)

d(OMk).

nik

lL

Proof. We know from Theorem 17 that the divergence and its inner product with the volume form are related by the formula

div(V) dMk = d(V J dMk) . A straightforward application of Stokes' theorem implies

J div(V)dMk = lk

J

d(V i dMk) = f V nl k

k

Let x E 8Mk be a point of the boundary. We decompose the vector V(x) into one part that is proportional to the exterior normal vector, and a vector W(x) that belongs to the tangent space T,ZBMk to the boundary:

V(x) = (V(x), N(x)) N(x) + W(x) . Moreover, note that the restriction of the inner product W J dMk to the boundary 8Mk vanishes identically. This implies that for the inner product of V with the volume form dMk. on the boundary 8Mk

V _j dMk = (V, N) N j dMk + W i dMk = (V, Al) N(x) _j dMk . Hence, we obtainJ

div(V)dMk = ik

f

Mk

(V, N) Ni dll'ik .

By Theorem 19, the inner product N(x) . dMk coincides with the volume form of the boundary, d(8Jik). 0 As a direct application of the Ostrogradski formula we obtain Gauss' theorem.

Theorem 27 (Gauss' Theorem). Let V be a vector field, let f be a function on the oriented, compact manifold Mk, and let N be the exterior unit normal vector field of the boundary. Then

J l (V, grad(f )) dMk + f l

k

f . div(V)dMk = J

alk f

(V, N) d(81bik) .

3. Vector Analysis on Manifolds

84

Proof. This equation immediately follows from the Ostrogradski formula together with the rule from Theorem 11

div(f V) = f div(V) + V(f) = f div(V) + (V, grad(f)) . In a similar way we derive Green's formulas in versions that are not confined to 12. First we generalize Green's first formula.

Theorem 28 (Green's First Formula). Let f, g : Mk

R be smooth func-

tions on the compact, oriented manifold Mk. Then

f

f'O(g) dMk +

g'ad(g)) dMk =

ff

(grad(.9)N) d(8Mk).

aMk

A1k

Proof. By the definition of the Laplacian, we have

f O(g) dMk = J %rk

f

'k f div(g'ad(g)) dAik .

Now apply Gauss' theorem. Applying Green's first formula twice leads to Green's second formula.

Theorem 29 (Green's Second Formula). Let f, g : Mk - III be two smooth functions. Then 1 [g.

(f)-f. (g)] dMk = f

N) ] d(aMk).

aMk

Ark

1-1

Remark. The scalar product (grad(f ), N) defined only on the boundary is

iaN aN

often denoted by the symbol Of ION, since it is the derivative of the function

f in the direction of the exterior normal vector. This leads to a different formulation of Green's second formula:

f [g

(f) - f . %(g)] dMk =

.

fMk

[g.-f.

]

Corollary 1. Let Mk be a compact, oriented manifold without boundary. Then (1)

f

div(V) dMk = 0 for every vector field V;

Mk

(2)

f go(f) dMk = f fo(g) dMk = - f (grad(f ). grad(g)) Alk

Mk

dMk

Mk

for any two functions f, g E CO°(Mk).

0

3.8. The Classical Integral Formulas

85

Hence the Laplacian is symmetric with respect to the L2-scalar product. Moreover, the choice of sign we adopted implies that it is non-positive. Corollary 2 (Hopf's Theorem). Let Mk be a compact, connected, oriented manifold without boundary and assume that the function f : Mk R sat-

isfies at each point the condition 0(f)(x) > 0. Then the function f is constant.

Proof. Integrating the assumption s(f)(x) > 0 over Mk and applying the symmetry of the Laplacian just proved, we first obtain

0< f 1-o(f)-dMk = J f-j(1)-dMk = 0, Mk fk

i.e., the Laplacian of f vanishes identically, A(f) = 0. Inserting f = g into Green's first formula (Theorem 28) and taking into account that the boundary integral vanishes by the assumption concerning Mk, this implies

J

grad(f)I2-dMk

= -Jntkf

0,

and hence grad(f) = 0. Thus f is constant, since Mk is connected.

0

Concluding this section, we formulate Stokes' theorem in its classical form on 1R3. Contrary to the preceding theorems involving the generalizations of divergence and gradient to manifolds as introduced in the second section of this chapter, this only involves the notion of curl on open subsets of 1R3 from §2.3.

Theorem 30 (Stokes' Theorem-Classical Version). Let M2 C R3 be a compact, oriented surface, let V be a vector field defined on an open subset M2 C U C 1R3, let N : M2 S2 be the exterior unit normal field to the surface M2, and let T : aM2 - T(aM2) be the unit tangent vector field on the boundary curve aM2 with the induced orientation. Then

f (curl(V), N) dM2 = f MZ (V, T) d(0M2). Proof. Consider the 1-form 4 := V1dx1 + V2dx2 + V3dx3

on U associated with the vector field V = V la/axl + V2a/axe + Via/ax3 as explained in §2.3 and its derivative

d`4 =

av2

aV1

axl - axe

,

I dx ndx2 +

, aV3 aV2 J dx2 ndx 3. ax, - avl ] dx ndx3 + [ ax3 ax2 - ax3

aV3

3. Vector Analysis on Manifolds

86

Recall that the curl of V corresponds to the 1-form *dwv. If, on the other hand. It : W -+ M2 is a parametrization of the surface with components h', h2, h3 and coordinates yl, y2 from W, then for the exterior normal vector N to the surface we have the relation 09

h x 09h

yl

/ II A

y2

09y1

y2 x ah

ll

Two arbitrary vectors v, w E R3 satisfy the identity 1Iv

x wII2 = det

(V, V)

(v, w)

(v, w) 1 (w, w) J

and hence the preceding equation implies

f.

x

ayl aye II For the first component of the normal vector N written in the coordinates II

y1, y2 this reads, e.g., as

N'dM2 = N' f dy' n dye =

ahe Oh3 _ah2 ah3 , dy' A [ay' aye aye ayl

1

dye .

On the other hand, it is easy to compute the pullback by h of the forms dx2 and dx3:

h*(dx2) =

OhY12

+ A2dy2, h`(dx3) =

ldy' + Oh3dy2.

A direct comparison implies the following formula, which is independent of the coordinates y', y2:

N'dM2 = dx2 A dx3 . Similarly one proves

N2dM2 =

- dx' A dx3,

N3dM2 = dx' A dx2 .

The scalar product of the curl of V with the unit normal field N multiplied by the volume form dM2 is thus simply the differential of wv: (curl(V), N) dM2

aV' 2 = [ aV3 ] N dM + 109x3 axe 09x3 = L aV2

1

dwv.

aV2

9V3

09x1, N2 dM2 + 09x1 - axe, N 3dltil 2 L aV'

Therefore, Stokes' theorem can be applied in the format

J (curl(V),N) dM2 = %f2

J Af2

&4 =

y=J[V1dx1 8M2

8M2

+ V2dx2 + V3dx3].

3.9. The Lie Derivative and the Interpretation of the Divergence

87

If, however, T is the unit tangent vector field to the curve 9M2, then d(8M2)(T) = 1, and hence

T'd(aM2) = dxl, T2d(8M2) = dx2, T3d(8M2) = dx3. Now we can rewrite the line integral above as

J [V ldxl + V2dx2 +V 3dX3]

= J1L12

(V,

T) d(8M2),

and, summarizing, we arrive at Stokes' classical integral formula.

0

3.9. The Lie Derivative and the Interpretation of the Divergence The aim of this section is to interpret the divergence of a vector field geometrically as the infinitesimal volume distortion of its flow. First we recall some results from the local theory of ordinary differential equations and introduce the flow on a manifold as well as the Lie derivative of forms. Then we compute this Lie derivative by means of the exterior derivative, which, in a special case, leads to the interpretation of the divergence mentioned in the title.

Let V be a vector field on the manifold Mk. An integral curve of V is a Mk whose tangent vector -y(t) = ry,(8/8t) at each point curve ry : (a, b) coincides with the value of the vector field there:

?(t) = V('Y(t)) The well-known existence theorem for autonomous differential equations states that for every initial point x E Mk there exists a maximal integral

3. Vector Analysis on Manifolds

88

curve ryx

:

(ax, bx) - Mk defined on an interval, containing the number

0 E R, satisfying

'Y.,(0) = X. Moreover, this maximal integral curve is uniquely determined by the initial condition. Denote by EV the set

EV = {(t,x)E]RxMk:ax
Mk defined by the formula

,Dt(x) = '(t,x) := 7x(t) the flow of the vector field V . If the maximal integral curves are defined for all values t E R of the parameter (i.e., if Ev = R x Mk), then the vector field V is called complete. A complete vector field V determines a one-parameter group 4Pt : Mk ---+ Mk of diffeomorphisms from the manifold to itself: 4)to+tl = 4t0 O 46tl . This relation between diffeomorphisms is a direct consequence of the uniqueness of the integral curve for a fixed initial condition. Conversely, if a oneparameter group of difeomorphisms is given, then we obtain a vector field

by looking at the tangents to the trajectories +t(7) of a point: V (X)

dt (bt(x))1t=o

Example 32. The one-parameter transformation group in R2 of the vector field

V = -x2871 +x1872 is determined by the differential equations

i1 = -x2

th2

= x1

and coincides with the group of all rotations in the plane.

Example 33. The one-parameter transformation group of the vector field V

= x1871

+...+x"ban

in ]R" is determined by the differential equations .i1 = x1,

...,

in = 7n

and coincides with the group of all dilatations in Ilt".

Suppose that the manifold Mk is a closed subset of R. Then Mk is a complete metric space: Every Cauchy sequence in Mk converges. For bounded vector fields we will prove

3.9. The Lie Derivative and the Interpretation of the Divergence

89

Theorem 31. Every bounded vector field on a complete manifold without boundary is a complete vector field. In particular, every vector field on a compact manifold without boundary is complete.

Proof. Denote the maximal length of the vector field V by K,

K := sup{IIV(x)JI: xEMk}. Let : (a, b) , Mk be a maximal integral curve of the vector field. and assume that b < oc. We apply the mean value theorem of differential calculus

to the vector-valued function y : (a, b) - AIk C IR". It states that for every two values of the parameter, a < t1 < t2 < b, there exists one more value,

tl < t' < t2, such that Ily(tl)-y(t2)II

<_

It2-tll Ily(t*)II = It2-tll'IIV(y(t'))II <-

Since b is finite, any sequence t; E (a, b) converging to b is a Cauchy sequence

in R. The last estimate shows that the image sequence -y(ti) E Mk is also a Cauchy sequence. By assumption, Mk is a complete manifold, and hence the sequence -y(ti) converges in Mk. This observation shows that the left-hand limit, lim y(t) _: x`, tb

exists. The point x` E Mk is an inner point of the manifold, since there are no boundary points by assumption. We choose a chart h : Rk -+ Mk around the point x' = h(O) and represent the vector field V in the corresponding coordinates,

V = Vla/eyl + ... + Vka/Oyk . The solution y(t) of the differential equations

yk = V k(yl, ... , yk) y1 = V 1(yl, .. . yk), .. converges for t - b to the point 0 E Rk. Hence the integral curve y(t) can be extended beyond the parameter b. This contradicts the maximality of b ,

0

resulting from the assumption b < oc.

Now we turn to the Lie derivative of a differential form with respect to a vector field. If w' is an i-form, then. pulling it back by the flow of the vector field, we obtain a 1-parameter family of forms, 4 (w'). The derivative of that

with respect to the parameter at t = 0 is the so-called Lie derivative. This concept involves the flow of the vector field only in a small neighborhood of zero and is thus well-defined for arbitrary vector fields.

Definition 20. The Lie derivative CV (w') of an i-form w' is the form CV(w`)

wt li o

3. Vector Analysis on Manifolds

90

Theorem 32. The Lie derivative of a differential form is expressed in terms of the exterior derivative and the inner product as

'CV (d) = d(VJ w')+VJ (dw`) Proof. The exterior product w' A 171 is bilinear, and the product rule is proved as usual,

Cv(w' Ar?) = Cv(w') AT? +W' ACv(T?) Furthermore, the exterior derivative d commutes with the induced maps 4)i, and hence we have

Cv(dw') = d(Cv(w)) Denote the right-hand side of the equation to be proved by C4(J). Then Cv(dw') = d(V J dw') + V J (ddwl) = d(V J dw')

= d(VJdw'+d(VJw')) = d(C*(w')). The first, purely algebraic equation,

VJ (w'Ar7') _ (VJ w')Ar7+(-1)'w'A(VJ r'), leads to the identity

Cv(w'Ar7') = d(VJ (w'Arl'))+VJ d(w'Ar1')

= d((VJ w')Aq'+(-1)'w'A(VJ r?))+VJ (dw'Ar7'+(-1)'w'Adr7') = d(VJ w')Ar7'+(-1)'-'(VJw')Adr7'+(-1)'dw'A(VJ r)) +w'Ad(VJ +(-1)'(V J w') A dr7' + w' A (V J dri)

= (d(VJ w')+VJ

r7')+VJ dr7')

= Lv(w')Arl'+w'ACv(r') Lastly. C and C coincide for functions,

Cv(f) = V(f) = VJ df = G`v(f),

0

and thus the formula is proved.

We apply the formula of the preceding theorem to the volume form of an oriented manifold Alk. From the definition of the divergence of a vector field we obtain Cv(dMk) = d(V J dAlk) = div(V) dAfk. Fixing a point x E Afk and choosing a neighborhood U(x, e) C Mk of radius > 0, we obtain by integration EJiv(V).dM' 1

= (l,e)

I l= /

3.9. The Lie Derivative and the Interpretation of the Divergence

91

Dividing both sides by the volume of the set U(x, e) and taking the limit for E - 0. we see that the mean value theorem of integral calculus immediately implies the formula

div(V)(x) = lim

vol(U(x, r))

(vol(t(U(x.e))))

.

u=o

This formula allows a geometric interpretation of the divergence of a vector field: It is the volume distortion of its flow in infinitesimally small time and on infinitesimally small domains surrounding the point x. Corollary 3. The divergence of a vector field vanishes if and only if its flow 4t : AIk _ Mk consists of volume-preserving diffeomorphisms.

Proof. For a measurable set A C Afk we denote by hA(t) := vol((Dt(A)) the volume of the set -(Dt(A). The derivative of the volume change is computed as follows:

d (hA(t 61t

=

dtd

I

f

fd(A)

dAfk)

=dt =

L1A)

Gv(dMk) = 41

div(V) dAlk. (A)

t (4)

From this we see that the flow 4bt preserves the volume if and only if the divergence of the vector field V vanishes identically.

0

The Lie derivative of a vector field W with respect to another vector field V is defined just like the Lie derivative for forms; nevertheless, it is called the commutator of both vector fields.

Definition 21. Let V and W be two vector fields, and denote by

the flow of the vector field V. The commutator [V, WI is defined as the vector field

[V, W1 = lim

t-0

W t

We prove an important formula relating the commutator of two vector fields to the exterior derivative of 1-forms.

Theorem 33. Let wl be a 1-form, and let V, W be two vector fields. Then dwl(V,W) = V(wl(W)) - W(w'(V)) -w'([V, WI)

Here V(wl(W)) is the derivative of the function w1(W) in the direction of the vector field V.

3. Vector Analysis on Manifolds

92

Proof. If x E Mk is a fixed point, then by the definition of the commutator [V, W]

x))) - W(x) .

[V, W](x) = Iio

The formula to be proved then follows from Theorem 32 by means of the following computation:

w'(W(x))

wl([V W])(x) = lim

t--o

'

=

li

o

t

(4'-t)'(f') - w'

(W(4'c(x))) + u ow'

(W(t(x)) - W(x))l

_ -(Cvw')(W)(x) + V(w'(W))(x) _ -W(w'(V))(x) - dw'(V,W)(x)+V(w'(W))(x).

0

If, in particular, wl is a closed form, then we obtain

w'([V, W]) = V(w'(W)) - W(w'(V)) In local coordinates the vector fields are represented by their components,

V=via k

Applying the above formula to the closed form dyt gives us k

dy'([V,W]) _

V;aWl

i=1

-

W1.

k i==11

This leads to the following local expression for the commutator of two vector fields:

k

[V , W]

_

(Viewt

J \1

1

- Wi avt) l a ay'

I ay'

Example 34. The commutator of the vector fields V = y a/ax, W = 0/4 defined on 1R2 is the vector field [V, W] = -8/0x. The algebraic properties of the commutator of vector fields are summarized in the following theorem.

Theorem 34. Let U, V, W be vector fields, and let f be a smooth function on the manifold Mk. Then the following identities hold: (1) [U + V, W] _ [U, W] + [V, W];

(2) [V, W] _ -[W, V]; (3) [U, [V, W]] + [V, [W.U]] + [W, [U, VJ] = 0 (Jacobi identity);

(4) [f V,W] =f

3.9. The Lie Derivative and the Interpretation of the Divergence

93

Proof. These formulas immediately result from the local expression for the commutator. We present the calculation for the last of the formulas:

taw,aw f-VI)l a (i.v' ayt ayt J' -l k

vi

[f v, w] _ E

wi av

vi awl

f 0=1

ayi

a

_

ayi ) ay,

-

(F ay, k of_) ((vi) E a wt i=1

1=1

OV

= In the following sense, the commutator is a natural operation between vector fields.

Theorem 35. For a diffeomorphism 4' : Mk -+ Mk and vector fields V, W on the manifold Mk the differential (P. is compatible with the commutator: 'F, ([V, W]) _

Proof. Note first that for every 1-form w' the functions 4'(w')(W) and w' ('. (W)) are related by the formula

'DO(wl)(w) = w'(=(w)) 04-'

-

Theorem 33 then implies wl ((I).[V+w]) = ('F*wl) ([V, W]) O'D

= V ((''wl)(W)) o 4)- W o 4)- d(4)`wl)(V, W) o 4) _ ('F.V) (w'(4'.W)) - ('.w) (U" (4%v)) - (dw') ('.v, 4.W) = w' (['.(V),4'.(W)]) This equation holds for every 1-form w', and hence the theorem follows.

Remark. Requiring that 'F be a diffeomorphism is often too strong. The compatibility of the commutator with the differential can sometimes be proved under weaker assumptions. For example, if the map 4) : M - N is smooth, the vector field V on M is said to be -t-related to the vector field W on N if at each point m E M $.,m (V(m)) = W('F(m)). If one can now show that, together with the pairs (VI, WI) and (V2, W2), the commutator pair ([V1, V2], [WI, W2]) is necessarily also fi-related, the same compatibility property follows immediately. To see this, apply the

3. Vector Analysis on Manifolds

94

commutator [WI, W2] to a function g : N --+ R: [WI, W2]$(m)(9) = WI(4i(m))(W29) - W2 (4)(m)) (WI 9)

= 4'.,m(VI(m))(W29) - 4)4,m(V2(m))(WI9) = VI (M) ((W2 9) 0 4') - V2 (M) ((WI 9) 0 4?).

On the other hand, (W1 g) o 4? = Vi(g o 4?), since

V+(n)(9 o 41) = 41.,n(Vi(n))(9) = Wi(4)(n))(9) = (Wi 9)(4'(n)) Thus we obtain [W1, W24(m)(9) = [Vl, V2]m(9 0 4>) = 4).,m[Vl, V2]m(9)

,

i.e., the vector fields [WI, W2] and [VI, V2] are, in fact, 4)-related.

The commutator of two vector fields measures the extent to which their flows do or do not commute. This explains the name for the vector field [V, W].

Theorem 36. Let V and W be two complete vector fields on the manifold Mk and denote by 4't and X8, respectively, their flows. Then the commutator [V, W] vanishes if and only if 4it o 4' = %P, o tt for all -oo < s, t < oo.

Proof. Because d

= tim (4'-h-h).W - (4'-t1) / = (4'-t1).([V, W]),

the commutator [V, W] vanishes if and only if the vector field W is invariant under the flow 4it, (4)1).W = W. This condition is in turn equivalent to the commutativity 4't o T, = T. o 4it of the diffeomorphism 4't with the flow ID, of W.

3.10. Harmonic Functions A function f : Mk - R is called harmonic if it is a solution of the homogeneous Laplace equation 0(f) = 0. As a special case of Hopf's theorem (Corollary 2), we have the following:

Theorem 37. Every harmonic function on a compact, connected, and oriented manifold without boundary is constant. If the boundary of the manifold Mk is not empty, there exist two particularly important boundary value problems for harmonic functions.

3.10. Harmonic Functions

05

The Dirichlet Problem: Assume that a function V : OMk - IR is given. We IR whose values on the boundary look for a harmonic function f : Mk aMk coincide with those of cp:

0(f) = 0 in Mk and

f IaaMk = cp .

The Neumann Problem: Assume that a function cp : OMk -+ IR is given. We R whose normal derivative on the boundary coincides with yp:

look for a harmonic function f : Aik

0(f) = 0 in Mk and eiv

= cp on aMk.

A solution of the Neumann problem is never unique. For each solution f the sum f + C is, for an arbitrary constant C, a solution of this problem, too. This is the only degree of freedom, since we have

Theorem 38. Let Mk be a compact, connected, and oriented manifold in R", and let cp : aMk -+ IR be a smooth function.

(1) The Dirichlet problem has at most one solution.

(2) If f,, f2 both are solutions of the Neumann problem, then f, - f2 is constant.

(3) The vanishing of the mean value of cp is a necessary condition for the solvability of the Neumann problem: JOMk

p d(aMk) = 0 .

Proof. If fl, f2 both are solutions of the Dirichlet problem, then the difference u := f, - f2 satisfies the equations

0(u) = 0

and u IBMk = 0.

From the first Green formula we obtain

0 = J fk

-Lk

U. 85Jk

a" aN Nd(aMk),

and hence the gradient of u vanishes. Since we have u IBMk= 0, the function u has to vanish identically. In the case of the Neumann problem the argument runs along the same lines. i.e., starting from the equations

0(u) = 0

and

au

= 0 on aMk

and the Green formula, we again conclude that grad(u) = 0. Thus the difference u = fl - f2 is constant. If the Neumann problem has at least one

3. Vector Analysis on Manifolds

96

solution for any given function cp :8Mk integral formulas, we obtain 0

R, then, applying the classical

r L . d(Wk) = J

f O(f) dMA =

d(OMk).

Mk

160

.

The Dirichlet and the Neumann problems have, for a given boundary condition cp : 8Mk R (satisfying f cp = 0 in the case of the Neumann problem) a solution. We will not prove this existence result here for general mani-

folds, but confine ourselves to the case of the ball D"(R) C R' of radius R > 0. For these spaces, there is an explicit classical solution formula which will be derived here. For the sake of simplicity we only discuss the case of

dimensions n > 3 and leave it to the reader to complete the discussion in dimension n = 2, which differs only slightly from the one below. We start with a few preparations. By

r=

(x1)2 +

... + (xn)2

we denote the distance from the point (x', ... , x") E Rn to the point 0 E Rn.

Lemma 3. Let u(x) be a harmonic function defined on the set Rn - {0}, and assume that u depends only on the radius r. Then there exist constants C1 and C2 such that C2

u(x) = C1 + -n-2 Proof. By assumption there exists a smooth function h : (0, oo) -' R satisfying u(x) = h(r(x)). Differentiating this, we obtain the formulas n( )(' i( )r2 - (xi)2 x' Ou OZU

h(r)r, ax = h

112

r \r/

+h

r r3 8x and the latter implies the following differential equation for the function h(r): _

n

0

O(u)

52x1

lh(r).

=

For n > 3 the function h(r) = C1 + C2 r2-n is the solution of this ordinary differential equation.

Let y E Rn be a fixed point. The function u(x) = JJx - y112-n defined on the set Rn - {y} is a translate of r2-n, and hence a harmonic function. On the other hand, the partial derivatives of a harmonic function are harmonic functions as well. Thus

u(x) - 2 .=1

au y ax'

I Ix112

Ilx

I lyl l2

yl In

3.10. Harmonic Functions

97

is a harmonic function defined on iRn - {y}. We will use this family of harmonic functions in the observations to follow.

Lemma 4. Let IIxii < 1 be any point in the interior of the unit ball D". Then

I

IIXII" n

1

IIx

dSn-1(y) = vol(Sn-').

Proof. First we prove that the function

f n-' IIx_ IIyIIn 2

h(x)

.

dSni-1(J)

depends only on the radius r = r(x). In fact, for a linear orthogonal map T : 1R" -+ R" we obtain, from I det(DT)I = I det(T) I = 1 and the corresponding transformation formula for the volume form of the sphere,

T*(dSn-1) = T*(NJ

d1Rn)

= NJ (T'(d1Rn)) = NJ dRn =

dSn-1,

the property to be proved:

h(Tx) = J _ I

dSn-1(y) n-1 IITxIITyIIn

1- IIXII2

= J n-s lixl

TII'(Iy)Iin

dSn-1(y)

. dSn-1(z) = h(x).

Jsn-' IIx - zlln If, in addition, h(x) is a harmonic function, then

z (h)(x)

IS, f

A. (1_ilx2

dSn-1(y))

= 0.

= yIIn The first lemma implies that there are constants C1, C2 with h(x) = C1 + C2 r2-". At the point x = 0 the function h(x) is a regular function and satisfies h(O) = vol(Sn-1). Thus the constant C2 vanishes, and C1 is equal to vol(Sn-'). 1

The solution of the Dirichlet problem for the unit ball D' C ]R" together with an explicit formula are the subjects of the next theorem.

12 is.,

Theorem 39. For every continuous function cp : S"-1 - 1R, the formula _ 1 yII" cp(y) dS"-'(y) AX) = vol(Sn-1) II-

IIx

defines a harmonic function in the interior of the unit ball, and at a boundary point z E Sn-1 lim f (t z) = cp(z).

t-1-

3. Vector Analysis on Manifolds

98

is harmonic with respect to the Proof. The function (1 - IIxII2)IIx variable z, and hence the function f (x) is harmonic, too. We prove that f(z) coincides with ;p(z) on the boundary Sn-1 in the way stated above. y E Sn-1} the maximum of the modulus of p Denote by in := and fix a positive number e > 0. The continuous function p is uniformly continuous on the compact set Sn-'. Hence there exists a number b > 0 such that for any two points y, z E Sn-1 in the sphere, i i - z < 6 implies yll-,,

the estimate I :p(y) - cp(z) I < e. We decompose the sphere Sn-1 = D1 u D2 into the parts

- zII > a} . D1 = {y E S" : IIy - ziI < b}, D2 = {y E Sn-1 For y E D2 and 0 < t < 1 we estimate the distance from y to the line :

II

segment between 0 and z E S"-': fly - tz1I2

=

1 - 2t(y,z) + t2 = (t - (y,z))2 + 1 - (y,z)2 > 1 - (y.

>

1 - (y, z) = 2-(2- 2(y, z)) = -Ily - zII2 >

1

1

Z)2

b2 .

2

We then split the difference _

P ' z) - Y (z) = vol(Sn-1) J$^

2

dSr

Y

Iltz

(?/)

into the integrals over D1 and D2. The modulus of the first integral can be increased to

1-IItzII2

dS"

n

1

1

S^-' Iltz - ylln Di We treat the second integral using the inequality obtained before:

D2

< 2m -

(1-t2)(

a2)n

vol(Sn-').

In summary, for every positive number E > 0 there exists a number 6 > 0 such that for all 0 < t < 1 the following inequality holds:

- t2).

E+2mb Hence the upper limit is bounded by E,

lim suplf(t' z) - V(z)I < E The last inequality holds for all positive numbers E > 0, and this in turn implies lim f (t tt-

z) = p(z)

.

0

99

3.10. Harmonic Functions

Next we will discuss several consequences of the solution formula for the IR defined on the Dirichlet problem. For a harmonic function f : D"(R) closed ball of radius R > 0, the function j (z) := f (R z) is also harmonic on the unit ball. Applying the previous theorem and returning to the vari-

able r = R z E D"(R) finally leads to the Poisson formula for harmonic functions: 2

2

-

2

f(x) = vo1Sn-1) J "-' IIR- RI yl l1"f(R y) dS"-1(y)

Evaluating the Poisson formula at the point x = 0 leads to Gauss' mea value theorem for harmonic functions.

Theorem 40. The value of a harmonic function f at the center of the ball coincides with the mean value of the harmonic function on the boundary of the ball:

We will use Gauss' mean value theorem in the proof of the maximum principle for harmonic functions.

Theorem 41. Every harmonic function f on the connected manifold Mn C R" attaining its maximal value in the interior of Mn is constant.

Proof. Denote by m the maximal value of f and by Q = {x E Al"\8AI" : f (x) = m} the set of all inner points of M" at which f attains this maximal value. By assumption 1 is a non-empty closed set in M"\811". Hence it suffices to prove that 1 is an open subset of M"\8M". Choose a point xo E S2 and a radius Ro such that the ball D"(xo, R0) with center x0 and radius Ra is completely contained in Mn\8M". By Gauss' mean value theorem

f (x0) =

VOl(S1

n-1)

Jsn_l

f (x0 + Ro y) dSn-1(y)

m. = .f (xo)

This implies that f is constant and equal to m on the sphere with center x0 and radius Ra. This observation can also be applied to each radius RR < Ro below R0. Together this implies that f - m is constant on the ball D"(x0i R0), i. e., D" (x0, Ra) is contained in Q. Thus Il is an open subset. 0

Eventually we will prove one more application of the Poisson formula, Liouville's theorem for harmonic functions.

Theorem 42. Every harmonic function f : (above) is constant.

1[P"

R bounded from below

3. Vector Analysis on Manifolds

100

Proof. Changing f, if necessary, by adding a constant, we can suppose without loss of generality that the function f is non-negative. Fix a point xo E Rn and choose the radius R such that xo lies in the ball D"(0, R). By the Poisson formula, Gauss' mean value theorem, and the assumption f > 0 we have Rn 2

f (xo) =

vol(Sn-1) Jsn

Rn-2

R' - II

2

IIo-R

yll In

R2 - I IxOI I2

< vol(Sn-1)

Jn-i IIIxoII - IIR

_

2 - II oIIRIn 2

2(

vo Sn-RIIlxoll

-2(R2 _

RSn

)IIIxoII

Isn

- iRI"

y) dS"-' (y)

. f (R

ylll

n

.f (R y)

dSn- ' (y)

f(R y) dSi-1(y)

f(0)

Taking the limit for R -* oo yields for all points xoi E R" the estimate

f(xo) <- f(0), and the maximum principle for harmonic functions implies that f is constant.

3.11. The Laplacian on Differential Forms This section is a supplement to the vector analysis on manifolds as explained

before. We will discuss the Laplacian acting on forms and its properties. To this end, we will start from an oriented manifold Mm of dimension m together with a family of non-degenerate scalar products

g=:TXM' xTIMR and assume that these scalar products depend smoothly on the point. This is understood to mean that for any two smooth vector fields V and W on M"' the function g(V, W) is smooth. Such a family {gx)xEAfm is called a pseudo-Riemannian metric on M. In general, it has to be neither positive definite nor the restriction of the scalar product on the ambient euclidean space.

In this situation the tangent space is a real oriented vector space. and we denote by k the index of the scalar product. According to Chapter 1, at each point there exist a volume form dMm(x) E A ,'(M') and a Hodge operator

* : A (Mm) - Ax'-'(Mm) ' Together we obtain an m-form dMm on the manifold, as well as a *-operator

associating with every i-form a smooth (m - i)-form. The algebraic rules

3.11. The Laplacian on Differential Forms

101

from Chapter 1 can still be applied. We will use these and Stokes' formula to determine the operator b : S i+1(Mm) -+ 1'l (Mm) adjoint to the exterior derivative d : S1'(MTh) - S1'11 (M') acting on i-forms. As is well-known, the adjoint operator is characterized by the requirement

J

(w',17'+1) . dMm =

!

(wi bi7i+1) . dMm .

f)fm

m

Here wi and rli+1 are forms with compact support disjoint from the boundary

of M', and (dwi, gi+1) denotes the scalar product induced on forms. By Theorem 5 in Chapter 1 we have (dwi

dMm = (-1)k&' A (*1]i+1) = (-1)kd(wi A (*,ni+1)) + (_1)k+i+lwi A (d * 17i+1)

Integrating this equation and once again applying the algebraic rules, we obtain (_I)k+i+1(-1)i(m-i)+k (dwi,,gi+') , dIim = wi A * * (d * 37i+l) Mm J fm (_1)$(m-i)+i+1+k (wi *d * qi+1) , dMm. = nMm

Thus we have proved

Theorem 43. The operator 6 adjoint to the exterior derivative d : S2i(Mm) --, 11i+1(Mm) on a pseudo-Riemannian manifold of index k is given by b(lli+1) = (-1)k+mi+l * d * ni+1

Definition 22. Let Mm be a pseudo-Riemanniani manifold. The HodgeLaplace operator acting on i-forms, A: S2i(M') 1'(M'), is defined by

°db+bd. Example 35. For a function wO = f, in the sense of this definition,

°f = bdf = (-1)k+l *d*df, and, in the case of a positive definite scalar product, ° coincides with the Laplace-Beltrami operator (see the remark following Theorem 11, §3.2). Considering, on the other hand, the pseudo-euclidean scalar product of signature (n, 1) in Rn+1, we obtain for the Hodge-Laplace operator the expression

°f =

i 2f

f

(92x1 +''' + a2xn

a2f a2xn+1

Rn+1 with this pseudo-Riemannian metric is called Minkowski space. Hence, its Laplacian on functions is the wave operator (also known as the d'Alembertian).

3. Vector Analysis on Manifolds

102

Definition 23. An i-form w' defined on a pseudo-Riemannian manifold is called harmonic if 0(w') = 0; it is called co-closed if bw' = 0.

In the following theorem we collect the properties of the Hodge-Laplace operator immediately resulting from the definition.

Theorem 44. (1) Let w', if be forms with compact support disjoint from the boundary of the manifold. Then

Im-

(A (u;'), rl')

dMm =

r

(w',0(rl')) dJV1-

(2) Aod=doA and Dob=boo. Finally, consider a compact manifold without boundary endowed with a Riemannian metric (k = 0). Then the second order differential equation 0(w') = 0 is equivalent to two first order differential equations. Theorem 45. Let MI be a compact, oriented manifold without boundary, endowed with a positive definite metric. Then the following conditions are equivalent:

(1) w' is a harmonic form, A(w') = 0; (2) w' is a closed and co-closed form, dw' = 0 = bw'.

Proof. If w' is harmonic, then

0=

r

J

f

f'^ 5fr J M'" This implies dw' = 0 = bw', since the metric is positive definite.

0

Corollary 4. Let Mm be a compact, oriented manifold without boundary, endowed with a positive definite Riemannian metric. If every closed 1 -form

on M' is exact, then there exist no non-trivial harmonic forms. Proof. In particular, a harmonic 1-form wl is closed, and hence wl can be represented as the differential of a function, wl = df. Since

0 = bwl = bdf = 0(f) , f is a harmonic function, and by Hopf's theorem (Corollary 2) it is constant on each connected component of M. Therefore, its differential vanishes,

w'=df=0.

0

Example 36. The assumption of this corollary is, for example, satisfied for manifolds Mm which can be represented as the union M' = U U V of two open sets U, V C M'", where

3.11. The Laplacian on Differential Forms

103

(1) the intersection U fl V is connected,

(2) U and V are diffeomorphic to star-shaped domains in R"'. In fact, Poincare's lemma implies the existence of two functions ft, : U - R

and fi : V

R such that wl = dfu over U and wl = dfv over V. On the

intersection we then have

d(fu - fv) = 0, and hence fuu - fi, = C is constant. Now

ifxE V. f(x) _ I fl(x)+C if x E U. fu (x) is a uniquely defined function on AP" satisfying wl = df. For n > 2 the sphere S' allows a decomposition of the required shape, and hence S" has no harmonic 1-form with respect to any Riemannian metric. Example 37. Consider the 2-dimensional torus in 1R'1. T2 = {(xi, s2, x3. xi) E 1R4 : (x1)2 + (x2)2 = 1,

and denote by i : T2

(x3)2 + (x4)2 = 1 }

R4 its embedding into 1R4. The 1-forms induced on

T2

:= i'(-x2dx1 + xldx2) and n1 := i*(-x4dx3 + x3dxa) are harmonic. To prove this, we use the parametrization h : [0,27r] x wl

[0.2a[ -+ T2 defined by the formula h(,jp, z') = (cos p. sin yp. cos i;'', sin r') .

In this chart,

h`(wl) = dp and h'(gl) = dv', and hence wl and 171 are closed forms on T2. Computing the coefficients of the Riemannian metric leads to the matrix 0

9iv

0

1

'

From this, it is easy to see that wi and ill correspond to each other via the Hodge operator of the torus T2: But then, wl and 91 are also co-closed, bwi = *d * wl = *dij' = 0. Harmonic forms play an essential role in topology and global differential geometry. Here we formulate the main result of Hodge theory; a proof would go beyond the scope of this elementary text. For a compact, oriented Riemannian manifold without boundary, (Mm. g), we introduce the following vector spaces:

3. Vector Analysis on Manifolds

104

(1) the space 7-l'(Mr,g) of all harmonic i-forms,

x' (M', g) = {w' : 0(w') = 0} ; (2) the space E(Mr) of exact differential forms,

E'(M') = {w' : there exists an (i -1)-form

n`-1 such that

d7ji-1

= w'} ;

(3) the space CE'(Mm, g) of co-exact differential forms CE' (M'", g) = {w' : there exists an (i + 1)-form tl'+1 such that 67i'+1 = w' }

Theorem 46 (Hodge's Theorem). (1) The space lf(Mm,g) is a finite-dimensional vector space. It is isomorphic to the de Rham cohomology HDR(Mm). For a fixed metric, each cohomology class of closed forms contains precisely one harmonic form. (2) With respect to the L2-scalar product, the spaces W, E' and CE' are orthogonal subspaces of the space Q'(M) of all square-integrable i-forms and decompose it into the sum

fl' = W ®E' ®CE'. From the Hodge decomposition we immediately obtain Helmholtz's theorem for 3-dimensional compact manifolds without boundary. Geometrically speaking, it states that each vector field can be represented as the sum of a gradient field, a curl field, and a harmonic field. In particular, the vector field is the sum of a curl-free and a divergence-free vector field. In the final chapter we will discuss and prove an analogous result for vector fields on R3 which decrease sufficiently rapidly at infinity within the framework of electrodynamics in detail. Corollary 5. For each vector field V on an oriented, compact, 3-dimensional manifold M3 without boundary, there exist a function f , a vector field W and a harmonic 1 -form ill such that wsd(1) + wcurl(W) + 171

In particular, wy is the sum of a closed and a co-closed 1 -form.

Proof. Applying Hodge's theorem to the 1-form associated with the vector field V, we conclude that there exist a function f, a 2-form (32 and a harmonic 1-form 771, together satisfying the equation

wv = df +0132+R1.

From §2.3, we know that the gradient of a function corresponds to the differential 4f,

df = wsd(I) 1

Exercises

105

Let W be the vector field corresponding to the 1-form *i3`, 2

1 =: WW

The curl of the vector field W is the vector field determined by the equation du) l

1

W = *wcurl(w)

But this implies

= * W = * * Wcurl(W) = wcurl(W) = *d * since in dimension m = 3 for a positive definite Riemannian metric the a2

I

1

1

relation ** = Id holds on the space of 1-forms. This completes the proof.

0

Exercises 1. a) The equation (x1)2 a12

+...+ (xann2

= 1,

where al, ...,a, > 0 are constants,

defines an (n - 1)-dimensional submanifold of R', the ellipsoid.

b) The equation x2 + y2 = r2 (r > 0 fixed) defines a two-dimensional submanifold of R3, the cylinder. c) Consider for every c E R the subset of R3 defined by

M, = {(x,y,z)ER3: x2+y2-z2=c}. For which values of the parameter c is Al, a two-dimensional submanifold of 1R3?

2. Prove or disprove the converse of Theorem 1: Let U C 1Rn be open, let

f : U - Rn-k be smooth, and denote by Al the zero-set of f, Al := {x E U : f (x) = 0}. If the estimate rk D f (x) < n - k holds for at least one point x E M, then M is not a submanifold.

3. Prove that the formula x2 + y2 = a2 cosh 2(z/a), a > 0 constant defines a two-dimensional submanifold of R3, which is, moreover, parametrized by

h(v',v2) = (acosh(vl/a)cosv2. acosh(v'/a)sinv2, vl) This surface is called the catenoid in differential geometry.

3. Vector Analysis on Manifolds

106

4. Let f : C

C be a complex polynomial f = Ek0 aiz` without double zeroes. Consider for every natural number I > 2 the set

M = {(z,w)EC2: w'- f(z) = 0). Prove that Al is a two-dimensional submanifold of C2 = ]R4. The assignment (z, w) w uniquely defines a map G : M C satisfying G'(z, w) = f (z). The set M is a so-called Riemann surface. The l-th root of f (z) is uniquely defined on it.

5. Consider the special linear group SL(2, R) of all real-valued (2 x 2) matrices of determinant 1: SL(2,1R) = JA E M2(IR) : det A= 1 }

.

Prove that SL(2, R) is a three-dimensional manifold, and that its tangent space at the neutral element E is determined by

TESL(2,R) = {AEM2(R): trA=0}. Hint: Apply Theorem 4 and the (well-known) formula det(exp A) =

etr.a

6. Prove that the orthogonal group O(n,1R) consisiting of all orthogonal (n x n) matrices

A A' = E }

O(n, R) = {A E

is a submanifold of 1Rn2, and compute its dimension. Is the subgroup SO(n.IR) of all orthogonal matrices of determinant I also a submanifold? If it is, compute its dimension, and describe its relation to O(n, R). 7. The set of all unit tangent vectors T1S2 to the sphere S2, T,S2 = {(x, v) E IR3 X 1R3 :

I

= IIvII =1, (x, v) = 0

} .

is a submanifold of R3 x 1R3 which is diffeomorphic to the manifold

{(z'.,

z 3)

E C3:

Iz1I2

+ Iz2I2 + 1Z312 = 1, (z')2 + (z2)2 + (z3)2 = 0 } .

8. The product of a manifold M without boundary with an arbitrary manifold N again is a manifold, and it has the boundary 8 (M x N) = Al x 8N.

9. The tangent bundle TM of every manifold M is always an orientable manifold.

Exercises

107

10. The topological boundary Fr(A) of a subset A C R" is the intersection of the closure of A with the closure of its complement,

Fr(A) := A n R' \A

.

Prove:

a) If M' C iR' is a closed m-dimensional submanifold of Rm, then OM' _ Fr(M"').

b) If MI C 1R" (m < n) is a submanifold with or without boundary of strictly smaller dimension, then always Fr(Mm) = Mm.

11. Find three vector fields of length one on the sphere S3 which are mutually othogonal at each point. 12. Construct a vector field of length one on each sphere of odd dimension.

13. Compute the representation of the following vector field in polar coordinates: V (X, y) = x2 + y2 (8/ax + 49/ay) . 14. Consider on the sphere S2 the vector field V (X' y, z) = ((x, y, z), (-y, x, 0))

Prove that V cannot be the gradient field of any function f : S2 which more general fact is this example based?

R. On

15. Compute the local coefficients (gij) of the Riemannian metric as well as the volume form of the following submanifolds in the respective charts: a) for the sphere S" in the chart determined by the stereographic projection h : 1R" -> S": 2 yi

h(y', ..., y")

( Jyi2 + 1'

2 y" ... +

1y12 - 1

JyJ2 + 1' 1y12 +

1)

,

b) for the pseudosphere, i.e., the surface of revolution defined by the tractrix in the parametrization

h(u',u2) = (asinu' cosu2, asinu'sinu2, a(cosu' +lntan(u'/2))) 16. Compute the formulas for the gradient of a function, for the divergence of a vector field, and for the Laplacian on the sphere S2 in the coordinates defined by stereographic projection.

3. Vector Analysis on manifolds

108

17. On an open subset of R3 (to be determined!) we introduce coordinates by the chart (rcosVcosv,rsin;pcos ,rsin;p)

on the parameter domain r E (0, oo), E [0, 27r], , E (-t'/2, o/2). Compute the coefficients of the Riemannian metric, and the formulas for the gradient, the divergence, and the Laplacian in these coordinates.

18. Let 111' and N' be k-dimensional manifolds. The map f : 111' , N' is called angle-preserving or conformal if, for two arbitrary vectors w1, w2 E

TZM' and their image vectors vi = (W;) E TjiyiNA, the respective angles coincide:

_

(y1, V2)

(wl, w2) 11w111. IIw211

]Iv1 11 -1111211

If, in particular, (h, U) is a chart of the manifold Alk, the chart h is anglepreserving if and only if the representation of the Riemannian metric in the corresponding coordinates is a multiple of the k-dimensional unit matrix E:

(gtj (y)) = A(y) . E,

,\(y) > 0.

19. Prove the orientability of the sphere S", using stereographic projection and Theorem 14.

20. Let AI' be a manifold, h : U _ M' a chart, and -y [a, b] --. h(U) C 111' a curve in Afk completely contained in the image set of h. Represent the curve 7 in the coordinates (h, U) as h-1 o ry(t) = (y1(t), ... , y'(t)). Prove that the length of the curve can be computed by means of the coefficients of the Riemannian metric g1, in the chart (h, U) via the formula :

b

l(7) = f a

1/2

9ti(7(t))d dtt) d dtt)

dl.

21. Let f : [a, b] , R+ be a positive function and let Ale = {(x. y, z) E R3 : y2 + z2 = f2(x)} be the corresponding surface of revolution in R3. Prove the volume formula /b

vol(M2) = 2n / f (x) 1 + (f'(x))2 dx. a

22. Compute the following surface integrals: for M2 = (x, y, z E R3) : x2+y2+:2 = a2. z > 01:

a) ty

-

Exercises

b)

109

/ (x2 + y2)

012.

hr

where M2 is the boundary of the subset of 1R3 x2 + y2 < z < 1.

described by the inequality

23. Compute the following surface integrals:

a) f2(R) (xdyAdz+ydzAdx+zdxAdy); [(y - z)dy A dz + (z - x)dz A dx + (x - y)dx A dy], where M2 is the

b) nJt2

boundary of the subset of R3 described by the inequalities x2 + y2 <

z2,0
24. Let the 1-form w1 on the Riemannian manifold Mk be given in the chart

h : V - Mk by wl = Ej f, dy'. Prove the following formula for the adjoint operator: k

b(w) =

1

a

t ay, (119-9*3fi)

25. Let Mk be a compact, oriented manifold without boundary. A function f : Mk -+ R is called an eigenfunction of the Laplacian with respect to the eigenvalue A E R, if 0 f= A f. Prove:

a) If fl and f2 are eigenfunctions with respect to different eigenvalues, 1\1

A2, then

(fl, f2) L2 = JM k fl(x)f2(x) dMk = 0. b) Let S2 C R3 be the sphere of radius 1,

AR3

the Laplacian on 1R3, A

the Laplacian on S2, f : 1R3 -+ R a smooth function, and set r = x2 + y2 + z2. Then (A`3

2

f)Is2 _ (arj +2

Of

+°S2(fIS2)

c) Let P : IIt3 -y 1R be a homogeneous polynomial of degree m which is harmonic, i.e. P satisfies OR'P(x) = 0. Prove that PIS2 is an eigenfunction of the Laplacian on S2, and determine the corresponding eigenvalue.

26. Let Mk C 1R" be a compact, oriented manifold, and assume that f Mk x [0, oo) -+ 1R is smooth. The heat equation is

ozf(x, t) =

af(x,

at

t)

3. Vector Analysis on Manifolds

110

Prove that if f is a solution of the heat equation satisfying f (x, 0) = 0 for all x E Mk and f (y, t) = 0 for all boundary points y E 8Mk and all times t E [0, to], then f vanishes identically on the set Mk x [0, to]. 27. Let Al"-1 be the boundary of a compact n-dimensional submanifold of 1R". and N(x) the exterior unit normal vector field. If V E 1R" is a constant vector, then

f(N(x), V) dM"-1 = 0. 28 . Prove the following commutator relations for the Lie derivative: [Cu, iw] = i(V,w]

and

[Gv, Lw] = G[v,wl .

Chapter 4

Pfaffian Systems

4.1. Geometric Distributions For smooth functions f 1, ... , f,"_k : R' - R with linearly independent differentials, the equations 1Vrk....,c,,,-k = {T E R' : J1 l2) = C1, ..., f,,,-k(X) = Cm_k I

define a smooth k-dimensional manifold. Linearizing this, in general nonlinear, system of equations by passing to the tangent bundle. we see that this manifold is described by the system Tblc,..... c,,,-k = {v E T lR"

:

df1(v) = 0, ... , dfm-k(v) = 0 1.

These equations determine at each point of IR"' a k-dimensional subspace of the tangent space to II2'. The resulting family of subspaces can also be described by different systems of 1-forms w1.... , wm-k. For example, if (hj) is a matrix of functions with nowhere vanishing determinant, the 1-forms

w; = E ik hj - dfj satisfy TMk.....cn,_k = {v E TIRt : w1 (v) = 0, ...,

0}.

The level surfaces, however, cannot be recovered from the knowledge of the forms w, alone. In general, the problem arises under which conditions linearly independent 1-forms wl, ... , wm-k describe a family of k-dimensional manifolds via the system of equations (Pfaffian system)

W1 = ... = wm-k = 0. Frobenius' theorem provides a complete answer to this question. This chap-

ter is devoted to its proof and the discussion of various applications. In 111

4. Pfafhan Systems

112

Chapter 5. we will apply it to prove the fundamental theorems of curve and surface theory.

Definition 1. A k-dimensional geometric distribution (or Pfaffian system) on Al'" is a family Ek = {Ek(x)} consisting of k-dimensional subspaces Ek(x) C Tj A1"' in the tangent spaces to Mm depending smoothly on the points in the following sense: For each point xo E M"'. there exist a neighborhood xo E U C Al' and vector fields VI, ..., Vk defined on U such that Ek(x) coincides with the linear hull of the vectors VI (x), .... Vk(x) at every point x E U.

Example 1. Every nowhere vanishing vector field V on AI' induces a onedimensional distribution El(x) formed by all multiples of the vector V(x). Conversely, every one-dimensional distribution El is locally determined by a nowhere vanishing vector field.

Example 2. The linearly independent 1-forms wl, ... , w,n_k on Al' determine a k-dimensional distribution by

Ek(x) = {vETAf' : wl(v)=...=W,,,_k(v)=0}. In analogy to the integral curve for a vector field, we now introduce the notion of an integral manifold for a distribution. Definition 2. Let Ek be a k-dimensional distribution on M'T'. A k-dimensional submanifold Nk C Mm is called an integral manifold of Ek if the tangent spaces of Nk coincide with the spaces of the distribution: T,XNk = Ek(x) for all x E Nk .

Definition 3. The k-dimensional distribution Ek is called integrable if it admits at least one integral manifold through each point x E Al'. Example 3. For a distribution Ek defined by linearly independent 1-forms W I , . . . , W,, -k as in Example 2, a submanifold i : Nk --+ Al"' is an integral manifold of Ek if and only if the restrictions of the forms to Nk vanish,

i*(wl) = ... = i*(wm-k) = 0. The local existence theorem for solutions of ordinary differential equations can be formulated as follows:

Proposition 1. Every one-dimensional geometric distribution is integrable. Example 4. Consider the nowhere vanishing 1-form w = x dy + dz on 1R3 together with the 2-dimensional distribution determined by w,

E2 = {vETR3: w(v)=0}. We show that this distribution is not integrable.

4.1. Geometric Distributions

113

Proof. Suppose that E2 is integrable. Then there have to exist an open R3 such that h'(W) = 0 and set W C 1R2 and a smooth map h : W rank (D(h)) _- 2. For example, one can choose for h a chart of the integral manifold. In the coordinates of R3, the map h = (h', h2, h3) consists of three functions, and the condition h`(W) = 0 is expressed on W C R2 by the equation

0 = h' dh2 + dh3 . Differentiating this expression and forming the exterior product with the 1-form dh2, we obtain

0 = dh' A dh2,

0 = dh2 A dh3 .

We then multiply 0 = h'dh2 + dh3 once again by dh' and take into account the equation we already derived, dh' A dh.2 = 0, to arrive at

dh' A dh3 = 0. In summary, all twofold products vanish, dhi Adhj = 0, and this contradicts

the assumption that the differential D(h) of the map h : W -. R3 has the maximal rank two. Hence the 2-dimensional distribution in 1R3 defined by w = x dy + dz cannot have an integral manifold. For higher-dimensional distributions on a manifold the problem thus arises: Under which conditions do they turn out to be integrable? The answer to this question forms the content of Frobenius' theorem. In order to formulate it, we also need the notion of an involutive distribution.

Definition 4. A distribution Ek on the manifold Mm is called involutive if, for every pair of vector fields V, Won Mm whose values V(x), W(x) E Ek(x) at each point belong to the distribution, the commutator [V, W](x) E Ek(x) again has values in Ek.

Theorem 1 (Frobenius's Theorem). Let Ek be a k-dimensional distribution on the manifold Mm defined by the (m - k) linearly independent 1 -forms W1.... , Wm-k: Ek

=

{VETMm: Wj(v)=...=Wm-k(V)=0}.

Then the following conditions are equivalent:

(1) Ek is integrable; (2) Ek is involutive; (3) for every point xo E Mm, there exist a neighborhood xo E U C Mm and 1 forms Oij defined on U such that m-k

dwi =

ei j A W j

j=1

4. Pfaffian Systems

114

for1
A Wm-k) = 0 Remark. The fourth condition occurring in Frobenius's theorem is called the integrability condition for the geometric distribution or the corresponding Pfaffian system.

Remark. A one-dimensional distribution is determined by m - 1 1-forms and then dWi n (Wl A ... A Wm-1) is an (m + 1)-form on the W1, .... m-dimensional manifold. This has to be zero for trivial reasons, and the integrability condition of Frobenius' theorem is automatically satisfied; see Proposition 1. We first prove some of the simpler equivalences in Frobenius' theorem and (4). proceed as follows: (2) (3) . (4) and the implication (1) The next section will be concerned with the proof of the central assertion of Frobenius' Theorem, i. e. the implication (3) (1).

Proof of the equivalence (3)

(4). Suppose that there exist local 1-

m-k

forms Oij such that dwi = E 9ij A wj. Then j=1

m-k dWi A (WI A ... AWr_k) _ E Oij AWj A (WI A ... A Wm_k) = 0, j=1

since the exterior square of any 1-form vanishes. Conversely, assume that condition (4) is satisfied. In a neighborhood U C MI of the point xo we extend the family of linearly independent 1-forms Wl, ... , Wm-k by adding 1-forms r1 j, ..., r1k so that the combined family, {wl, , Wm-k, 111, . . . , nk ), forms a basis f o r A (Mm) at each point x of U. The 2-form dwi (1 < i < m - k) can thus be represented as

m-k k k m-k dwi = E CQ0 WQ A Wp + E E Da j - Wa A nj + E Eji - rlj ^ nl 0=1 j=1

Q,3=1

j,1=1

with functions CQo, DQj and Ejl. The condition dwi A (W1 A ... A Wm_k) = 0 implies k

E Ejl-njA171A(W1A...AWm_k) = 0, jd=1

and hence the coefficients vanish, Ejl = 0. If we introduce the 1-forms m-k k 3=1

j=1

4.1. Geometric Distributions

115

the exterior derivative dwi takes the desired shape, m-k

dwi = E Bia n wa

.

11

a=1

Proof of the implication (3) (2). For any two vector fields V and W with values in the distribution £k, we have wi(V) = wi(W) = 0. From dwi(V. W) = V(wi(W)) - W(wi(V)) - wi([V. W]) we obtain

dwi(V,W) = -wi([V,W]). If now condition (3) holds for the distribution £k, then m-k

dwi(V,W) _ > Oji A wj(V, W) = 0. j=1

and hence all 1-forms w1, ... , w,,,_k vanish on the commutator [V, WJ. There-

fore, this vector field takes values in £k, i.e.. the distribution £k is involutive.

Proof of the implication (2) ; (4). Let £k be a k-dimensional involutive distribution. The form dwi A (wl n ... Aw,,,_k) has degree (m - k + 2). Inserting (m - k + 2) vector fields into this form, we can assume, without loss of generality, that two of these vector fields have values in £k. Denote them by V and W. Because of the involutivity of £k, we have

w;(V) = w;(W) = 0 and dwi(V,W) = -wi([V,W]) = 0. Therefore, the exterior product dwi A (wl A ... A wm-k) vanishes on every (m - k + 2)-tuple of vector fields.

Proof of the implication (1) : (4). The proof of this implication proceeds like the one before. Let £k be an integrable distribution, xo E All a fixed point, and h : W -+ All a parametrization of an integral manifold through this point. Then we have h*(wi) = 0 (1 < i < m - k), and this implies h`(dwi) = 0. Thus the 2-form dwi vanishes on the k-dimensional subspace £k(xo), dwi IEk(zo)XEk(zo)

= 0.

Inserting again (m - k + 2) vectors into the form dwi A (wi A ... Awm_k), we

conclude that at least two of these vectors-call them V and W-lie in the subspace Ek(xo). Hence wj (V) = wj (W) = 0 and &&,i (V, W) = 0. As above, we conclude that the form dwi A (wl A ... A Wm_k) = 0 vanishes.

4. Pfaffian Systems

116

4.2. The Proof of Frobenius' Theorem The implication (3) = (1), which is the core of Fobenius' theorem, is a local statement. Hence, without loss of generality, we can suppose that the manifold Mm is an open subset of R. We will prove a slightly more general result from which the proof of this implication immediately follows.

Theorem 2. Let w1i ... , w,,,_k be linearly independent 1 forms on an open subset M"' C Rm such that m-k

dwi = E B.ij A wj j=1

for certain 1 forms Oij. Then there exist at each point x E Mm a neighborhood U C Mm of x and functions hj and fj defined on the set U satisfying m-k

w= = j=1

Proof of the implication (3)

(1). Let £k be a distribution with the

property stated in condition (3). By Theorem 2, we can represent the forms w; in a neighborhood U of an arbitrary point zo E M' as m-k j=1

for certain functions. By assumption, the 1-forms w1, . . . , w,,,_k are linearly independent. Thus the differentials dfli ... , df,,,_k are linearly independent as well, and the set

Nk = (X E U : f1(x) = fi(xo), ..., fm-k(X) = fm-k(XO) } is a submanifold containing the point xo E M. At an arbitrary point x E Nk, we determine the tangent space:

TxNk = {v E TM' : df1(v) dfm_k(v) = 0 } C {v E TM' : w1(v) _ ... = w,,,-k(v) = 0) = Ek(x). For dimensional reasons, the vector spaces coincide, i.e., Nk is an integral manifold of the distribution £k through the point xo E Mm, and thus the integrability of the distribution £k is proved. 0

Proof of Theorem 2. We proceed in two steps. First we reduce the proof to the case of a system of 1-forms w1, ... , wm_k in a special normal form. To this end, we represent the euclidean space R n as a product RI = Rk X am-k and denote its points accordingly by y = (y', ... , yk) E Rk and z =

4.2. The Proof of Frobenius' Theorem

117

... , zm-k) E Rm-k. It is sufficient to prove the claim in Theorem 2 for 1-forms of the following special type:

(z1,

k

wi = dz`-1: Aii(y,z)'dy'

(1
j=1

In fact, representing the linearly independent 1-forms wi in coordinates, in

wi = E Pia da ° , a=1

we may first assume that the determinant of the part of the matrix (pij) containing the first (m - k) columns and rows does not vanish. Let (qjj) be the inverse of this matrix, and consider the 1-forms m-k m-k in in

w{ _

giaPa3d23 = dr'- E d..

gia'wa =

3=m-k+1

a=1 3=1

0=1

mk giapa3

3

0=1

Thus the forms w; are of the special type above. Moreover, m-k rn-k m-k m-k

dwi = E dgia A wa + a=1

m-k (M- k

dgia A wa + Y, qia ' Ba 3 A W3 a=1

a,3=1

m-k dgia + E gi03ae3ry

_EE a=1

qia . d41a = a=1

3=1

A wa,

3.y=1

and the system of forms wi, ... , wm-k also satisfies the condition m-k

dw; _

B a A UJ

a=1 . If Theorem 2 now holds for this system of forms, we obtain functions h,j and fj* such that

for certain 1-forms O

= m-k w;

> h,j

dfj

.

j=1

Rewriting this leads to m-k

Wt =

m-k m-k

Pia ' wa = E E 11aj ' Pia ' dfi a=1 j=1

a=1

Summarizing, it is sufficient to prove Theorem 2 for forms of the type k

wi = j=1

4. Pfafan Systems

118

Without loss of generality, we may choose the origin, x = 0 in Rm, as the point x = (y, z) in whose neighborhood we want to represent the forms w; in this special way. For fixed parameters (y', ...,y") E Rk near 0 E Rk, we consider the ordinary differential equation with initial conditions k

(t) _ > A;j(t - y, z(t)) yt,

z'(0) = z',

1 < i < m - k.

j=1

Denote by F` (t, y, z) the solution of this differential equation that is uniquely

determined near zero. These are m - k smooth functions in the variables t E R, y E 1Rk and z E ]R"'-k defined on a neighborhood of zero in R x RI. Moreover, for small values of the parameters we have F'(µ t, y, z) _ F'(t, p y, z), since the functions

G'(t,y,z) F'(µ't, y, z) satisfy the initial conditions G'(0, y, z) = z', and they are also solutions of the differential equation corresponding to the vector p y, k

Y, Z) = p'F"(p.t,y,z) _

A:,(lA't_

j=1

We introduce a coordinate transformation near zero in R- = ]Rk x ]R--k by means of the equations

u .= y,

F(1,u,v) := z.

The Jacobian of this transformation is ay,

a(u,v) zIo

0 E0lJ LE

((

since, by the differential equation and the initial condition,

=

az' I 0v j

= 8v'

OF' (1, O, v) I 8 .7

o

&0 0 =

0

.

This observation shows that both these equations determine a local diffeomorphism from the space RI to itself near zero. We represent the 1-forms WI, ... , wm-k in the {u, v}-coordinates: m-k

w; _

k

Bi j (u, v) dut + j:Pij (u, v) - duj j=1

.

j=1

Obviously it suffices to prove that the functions P;j - 0 vanish identically. First we derive the identity k

0. j=1

4.2. The Proof of Frobenius' Theorem

119

To this end, consider the map 4i : R' x Rk(t, y) -p Rk x R' -k(u, v) from the space R1 x Rk with coordinates (t, y) to the space Rk x R' -k with coordinates (u, v) defined for a fixed point v E R'"-k by the formula 40(t. y) := (ty, v). Then k

V (wi) = E Pig (t y, v)(t dyi + y dt) . j=1

On the other hand, with respect to the (y, z)-coordinates on R, the map 4i : R' x Rk(t, y) - Rk x R"'-k(y, z) is determined by

4'(t, y) = (t y, F(1, ty, v)) = (t y, F(t, y, v)) From the normal form, k

wi = dz' -

Aid

91,

j=1

we obtain for the induced form V(wi) the new expression

V (w;) =

OF'

at

k OF' dt+E-y j=1

k

j=1

Comparing in 4' (wi) the coefficient at dt and taking the differential equation into account leads to the identity k

k

.i=1

i=1

0= In the final step of the proof of Frobenius' theorem, we now show that the functions Pij = 0 vanish. To do so, we again resort to the map 4'(t, y, v) :_ (t y, v), but consider it this time as a map from R x Rk x RI-k 9 (t, y, v) to Rk x R°'-k 3 (u, v) (v is no longer assumed to be constant). The identity already proved implies the formula m-k

k

j=1

j=1

4)*(wi) = We denote by Pi*j the function

P*j(t. Y, v) := t Pip(t y, u). Then k

4 (wi) _ E P'j(t, y, v) dyt + (terms in dv'}, i=1

4. Pfafan Systems

120

and for the exterior derivative we obtain the expression

d-P`(wi) _

k 8P j=1

at

dt n dyj + {terms without dt}. m-k

By assumption, there exist 1-forms satisfying dwi = F, 9,1 A wj, and we j=1

represent these as V (9ij) := H,jdt + {terms containing dy', dvj}. This leads to a homogeneous differential equation for the functions we are interested in:

ps

m-k

E H.PQ> Q=1

with the initial condition P,j(0, y, v) = 0. The only solution of this homogeneous differential equation with the given initial condition is P;j (t, y, v) - 0, and this immediately implies Pij(t y, v) = 0. Thus the proof of Frobenius' theorem is completed. 0

4.3. Some Applications of Frobenius' Theorem The simplest case is that of an (m-1)-dimensional distribution £'"-1 on an m-dimensional manifold Mm. If Eii-1 is defined by one nowhere vanishing 1-form w, the integrability of the distribution reduces to the condition that the 3-form dw n w vanishes,

dwAw = 0. The method to explicitly integrate this (m - 1)-dimensional Pfaffian system is based on looking for a so-called integrating factor and an application of Poincare's lemma. Definition 5. An integrating factor for the 1-form w is a nowhere vanishing function f : Mm IR such that the 1-form f w is closed,

d(f . w) = 0. Theorem 3. Let w be a nowhere vanishing 1 form on the manifold Mm. (1) If there exists an integrating factor for w, then dw n w = 0. In this case the distribution Em-1 is integrable. (2) If dw Aw = 0, then there exists an integrating factor for the 1-form w in a neighborhood of each point in M'". (3) Locally, the integral manifolds of the distribution £'n-1 are the level surfaces of the function g determined from the integrating factor f via the equations

d(f w) = 0, f -w = dg.

4.3. Some Applications of Frobenius' Theorem

121

Proof. The equation d(f w) = 0 implies df A w + f dw = 0. Multiplying this equation once again by the 1-form w leads to f dw A w = 0. Since f # 0, we obtain dw A w = 0 as a necessary condition for the existence of an integrating factor. If, on the other hand, dw A w = 0, then the existence of an integrating factor follows immediately from Theorem 2, §4.2.

In dimension m = 2 the 3-form vanishes, dw A w = 0, for purely algebraic reasons. Thus Corollary 1. Every nowhere vanishing 1-form on a 2-dimensional manifold locally has an integrating factor.

Example 5. Consider in R2 the differential equation

P(t, X) + Q(t, x) i = 0. Near a point (to,xo) E 1122 at which P and Q do not vanish simultaneously, we have the 1-form

w= and its integrating factor f (t, x). The equivalent differential equation

(f P) (t, x) + (f Q) (t, x)± = 0 is called the total differential equation, and the solution curves are implicitly determined by the equation

g(t, x) = const with dg = f w. Frobenius' theorem now claims that it is always possible to solve the original differential equation by using the outlined method. It does not, however, provide an algorithm for finding the integrating factor. In simple cases this may be computed directly. If we can find, e. g., functions

F(t) and G(x) depending only on the variables t, x and satisfying OP(t, x) Ox

-

OQ(t, x) at

= Q(t, x)F(t) - P(t, x)G(x) ,

then f (t, x) = of F(t)dtef G(x)dx is an integrating factor. Example 6. Consider the differential equation (2t2 + 3tx - 4t)± + (3x - 2tx -

3x2) = 0.

We can choose F(t) = 2/t and G(x) = -5/x and obtain the integrating factor f = t2x-5. The solutions of this differential equation computed using this integrating factor are the curves described by the equation

t3x-4 - 2 t4x-4

- t3x-3 = coast

(compare the figure on the next page).

4. Pfaffian Systems

122

-0.2

-a .4

The curves t3 -

zt2

0.4

0

- tax = cx4 for different values of c

Assume now that on the manifold M'" we are given a Riemannian metric (, ) as well as an (m -1)-dimensional integrable distribution E'-1 described by the 1-form w. Denote the vector field associated with the 1-form via the Riemannian metric by W. This is uniquely determined by either of the two equivalent equations

*w = W j dM'

or w(V) = (V, W)

.

Using the vector field W, the distribution E'"-1 can be described as E"'_1

(v, W) = 0) ,

= {v E TM' :

and hence W is orthogonal to each integral manifold Nin-1 of the distribution. Normalizing the length of the vector field W to one, the volume form of each integral manifold is given by the formula dN'"-1

=

-

W i dM'" .

We study the behavior of the integral manifold N'"-1 C Mm of the distribution under the flow 4b:: M'" -i Mm of the vector field W. Compute first the Lie derivative of the 1-form w with respect to the associated vector field W. Since w(W) = 11W112, the formula £w(w) = d(W J w) + W i (dw) implies

(Gw(w))(V) = W(w(V)) - w([W, V))

4.3. Some Applications of Frobenius' Theorem

123

In particular, for a vector field V tangent. to the distribution this formula simplifies to

(,Cw(w))(V) = - w([W, V1)

(W, [W, V])

,

and we arrive at

Theorem 4. The flow 4it of the vector field W maps an integral manifold of the distribution E"`-1 to another integral manifold if and only if

V(IIW112)+dw(W,V) = -(W, [W,V]) = 0 for every vector field V on M with values in

Corollary 2. Let the distribution

£'"-1

Em-1 be defined by the closed 1 -form w.

Then the flow of the dual vector field W transforms integral manifolds into integral manifolds if the length I I W I is constant on every connected integral I

manifold.

Corollary 3. If the distribution E'"-1 is defined by a 1 form w of constant length. and if, moreover, the flow of the dual vector field W transforms integral manifolds into integral manifolds. then dw = 0. In this case 6'-1 locally consists of level surfaces of a function whose gradient has constant length.

Proof. First, Theorem 4 implies W . dw = 0. At a point x E Alm, we choose an orthogonal basis e1, ... , e,,, in the tangent space so that W is proportional to el . W = a el. Denoting by al, ... . o. the dual basis, the form w = a of is proportional to al. Represent the 2-form

du; = Eb;j o1Aa t<J

in this basis. Since 0 = dw A w, the 2-form dw only contains the in - 1 summands dw = b12 Q1 A (72 +... + bi 2al A a,,,. But then the condition

0 = W-i dw = a(b12a2+...+bjmam) means that w is a closed form. We compute the infinitesimal volume change of a compact integral manifold

N'"' 1 under the flow 4)t of the vector field W. The Lie derivative of the volume form dN1-1 is ,Cw(dA"-1) =

£w(III

.WJdAm) = WJd(IIWII.WJdMm)

_ (div(W) - 2 W(ln IIWII2)) and hence we obtain

dN'"-1

4. Pfaffian Systems

124

Theorem 5. The derivative of the volume change of a compact integral manifold N` of the distribution Em-' under the flow of the vector field W is given by the formula (vol(Ft(Nm-1)))

It=o

dt

=

f

(div(W) - 2 - W(ln IIWII2)) -

dNm-'

If, in particular, the distribution Em-' consists of the level surfaces of a function f : Mm -' IR, and we choose the gradient of this function as the vector field, W = grad(f), then div(W) = 0(f) is the Laplacian of f.

Corollary 4. The volume change of a compact level surface N` of the function f : Mm the formula

R under the flow of the gradient vector field is given by

d (Vol(4 (Nm-1))) 1, =0 = JN^

(°(f) - -grad(f) In ilgrad(f)I12) -dNm-'. 1

Remark. In all these formulas the Laplacian, the divergence, and the gradient are taken with respect to the manifold.

Example 7. Consider a function f : Mm - R and assume that there is another function µ : Mm -+ IR such that

d(Ilgrad(f)112) = 2p . df By Theorems 4 and 5, the flow of the gradient vector field grad(f) maps level surfaces of f to level surfaces, and the volume change is described by the formula Wt

(vol (41(Nm-1))) It=o =

J

vm

(0(f) -

µ)dNm-1

For example, the spheres S'-'(R) C R" are the level surfaces of the function f (x) = IIxII2, and we obtain Ilgrad(f )II2 = 411x112 as well as 0(f) = 2m.

Hence 0(f) - µ =_ 2(m - 1) is constant, and for the flow 4 (x) = e2t . x we obtain the following differential equation describing the evolution of the volume: wt-

The second application of Frobenius' theorem will play an important role in the chapter devoted to surface theory.

Theorem 6. Let 0 = (wig) be a (k x k) matrix of 1-forms defined on a neighborhood of 0 E Rm, and let AO be an invertible (k x k) matrix. In a connected neighborhood 0 E V there exists a (k x k) matrix A = (fib) of functions satisfying Il = dA - A-'

and A(O) = Ao

4.3. Some Applications of Frobenius' Theorem

125

if and only if d12 = 12 n Q.

In this case, the function matrix A is uniquely determined. If, in addition, 11 is an anti-symmetric matrix (12+12' = 0), and AO is an orthogonal matrix (Ao Aa = E), then the solution A(x) is also orthogonal at each point of the set V. Proof. First, the condition d12 = 1Z A 12 is necessary for the solvability of the equation Q = dA A-1. In fact, dA = SZ A implies

0 = ddA = d(f?-A) = (dIl)-A-I1AdA = The matrix A is invertible, and hence d12 = fZA11. Uniqueness of the solution

is also easy to see. For two solutions A(x) and B(x) we have, e. g.,

d(B-1) =

(dB) - B-',

-B-1

and hence the differential d(B-1 . A) vanishes: d(B-'

- A) = d(B-1) - A+ B-1dA =

-B-1

B-1

(dB)

_

A+ B-1 - 12 A

0.

Thus B-1 A is constant, and at the point x = 0 it is equal to the unit matrix. This implies A(x) = B(x) for all points x E V. Now we prove the existence of a solution under the condition d11 = 12 A Q. To this end, consider the following (k x k) matrix of 1-forms on the space Rm x IItk2 with coordinates (x1, ... , x"', z') ): k-

A= r=1

From d12 =12 A f? we obtain

dA = ddZ-d1ZAZ+flAdZ = -12A12AZ+SlAdZ = -1lAQAZ+1lA(A+12AZ) = fIAA. k

The system of forms dz'j - E Wjrzr3 is linearly independent in IR

x Rk2,

r=1

and by Frobenius' theorem, there exists an m-dimensional integral manifold Mm c IRt x Ht k2 through the point (0, A0) E Ht'" x Rk2. The tangent space T(o,Ao)Mm to this integral manifold has only the null vector in common with Rk2

TiO,AaiMm n

ittk2

= {0} .

This follows directly from the shape of the form A, since the tangent space to

Mm is determined by the equation A = 0. Then the integral manifold Mm

4. Pfaffian Systems

126

is the graph of a map A : W -+ Rk2 defined on an open set 0 E W C R'" satisfying the initial condition A(O) = Ao. From

A'(A) = we see that the (k x k) matrix A is the solution of the differential equation we looked for. The remaining statements of the theorem follow from the formula

d(At A) = (dA)t A + At dA = At 52t A + At

f2 - A

.

In fact, if f2 is an anti-symmetric matrix, and dA = f2 A is a solution of the differential equation, then we immediately conclude that d(At - A) = 0. O

Exercises 1. Consider on R3 the 1-form

w= Idx+Idy+1dz. yz xz xy Prove that the distribution defined by w = 0 is integrable. Find, moreover, an integrating factor for w, i.e. , a function f such that d(f w) = 0. 2. Let P and Q be functions in the variables x, y. u, v. Consider on RR the 1-forms

wi =

w-2 =

a) What are the conditions for the distribution wi = w2 = 0 to be integrable?

b) Let the function f : C2 --+ C be defined in the complex variables z x + iy, w := u + iv by f (z, w) = P + i Q. Prove that the integrability conditions for the system wi = w2 = 0 are, in particular, satisfied when f is a holomorphic function. 3. Prove that in R3, a distribution defined by the form

w=

P Q R det

NU

[PQRJ

=0

Exercises

127

vanishes. This formula is to be interpreted in the following way: Write the determinant out and apply each of the resulting differential operators to the respective function. 4. Solve the initial value problem i = ex cost, x(O) = xo a) by separation of variables; b) by finding an integrating factor.

5. The method of solving a differential equation by finding an integrating factor is of interest, in particular, if separation of variables is impossible. As an example, solve the differential equation i(x3 - tx) = 1, x(0) = xo. How could this equation, nevertheless, be solved by a suitable separation of variables?

6. Solve the differential equations a)

b) sin t + ez + cost i = 0 ; c) 4t3+6tx3 + (3t3x2 +3)i = 0

.

7. Let f = (w23) be a (k x k) matrix of 1-forms defined on an open neighborhood of 0 E R'. Prove that locally there exists a (k x k) matrix A consisting of functions satisfying

dA = if and only if there exists a 1-form a such that the matrix E := SZ - a E solves the equation dE = E A E .

Chapter 5

Curves and Surfaces in Euclidean 3-Space

5.1. Curves in Euclidean 3-Space The notion of a curve in 3-space R3 is not as simple as it may seem at first glance. For a long time in the 19-th century, the common understanding was that a curve is a subset A C 1[13 described in a continuous way by a real parameter. In this sense, a curve was the surjective image of a continuous

map f : [0, 1] - A C R3. To require the injectivity of f is too strong, in so far as it excluded curves with self-intersections. Apart from the fact that the assumption of continuity is too weak to render such a "curve" accessible to differential calculus. a much more essential reason shattered this notion of a curve.

Theorem 1 (Peano 1890). There exists a continuous. surjective map from the interval onto the square.

Proof. We construct the map f : [0, 1] - 10, 1] x [0, 1] as the limit of a uniformly converging sequence of continuous maps f,. In the first step, we divide the interval [0, 1] into nine subintervals of equal length and decompose the square accordingly. The map f, is then to map each of these subintervals, [(i - 1)/9, i/9], to the diagonal of the i-th square continuously and bijectively (see the figure). The map f2 arises in the same way when we further divide

each subinterval [(i - 1)/9,i/9] as well as each subsquare. This results in 81 intervals and squares, and f2 maps the small intervals continuously and bijectively onto the diagonals of the small squares 129

5. Curves and Surfaces in Euclidean 3-Space

130

7

1

0 1/9

8

9

5

4

2

3

9/9

so that f2([(i - 1)/9, i/9]) lies in the i-th square of the first step of the decomposition. By construction, f1(i/9) = f2(i/9). This way we obtain a sequence of continuous maps satisfying (1)

II

f.+1(t)II < V 2-13n for all t E [0, 11;

(2) f,,(i/3n+1) = fit+1(i/3n+1) = fn+2(Z/3n+1) for all n and i < 3n+1.

61

6

63

64 65

60

/5

78

80 81

78

71

73

7 76 7 75

3 ( 38

58

6 68

55

57

7

54

52

49

51

<

s

3 36

35 34

41

4

31

3 33

44

4

48

4

46

30

29 28

7

8

9

11

25

26 27

6

5

4

14

24

23 22

1

2

3

17

19

4

2

21

The estimate in (1) implies the Cauchy condition, 1

n (t) - rf (till < 11f - V/2- .3n-1'

for all t E [0, 1] and in > n. Thus, the sequence fn converges uniformly to a continuous map f : [0, 1] [0, 1] x 10, 1]. By construction, all the points 3^') belong to the image of the map f. The interval [0, 1] is compact, (s , and hence the image set f([0,1]) is again a compact subset of the square. But the subset of the image of f mentioned before is dense in the square

5.1. Curves in Euclidean 3-Space

131

[0, 1] x [0,1], and hence f ([0, 1]) = [0,1] x [0. 1], i.e. , f is continuous and surjective.

This example had an essential impact on general topology. In fact. it shows that the "dimension" of a topological space can increase under a continuous map, and that this notion has to be made more precise (topological dimension theory). We will not deal with this problem here, and instead confine ourselves to the case of smooth curves in R3. We could try to consider these as smooth, one-dimensional submanifolds of 1183, but that would exclude self-

intersections of curves again. For this reason, we formulate the notion of curve slightly more generally.

Definition 1. A (parametrized) curve is a differentiable map y' : [a, b] from an interval to 1183 whose derivative vanishes nowhere.

1<83

It is easy to see that sufficiently small image sets of a parametrized curve are submanifolds. Since dy(t)/dt 0 0, for a fixed initial point, the length function

L(t) := fat

dp.

II do (u) II

is monotonously increasing. A point on the curve is uniquely determined by the parameter describing the length of the curve segment from the initial point to the point considered. In other words, inverting the length 1R3 of the function, we obtain a parametrization -y o L-1 [0, L(y)] curve y, and the tangent vector with respect to the length parameters has length one, IIdy/dsIj = 1. This kind of parametrization is called the natural parametrization of the curve. The preceding observation shows that the natural parametrization can be constructed starting from an arbitrary one. We agree on the following: If the curve is given in any parametrization y(t), then we denote by y(t), etc., the derivative with respect to the parameter t; if the curve is given in its natural parametrization, then -y(s) is the derivative with respect to the length parameter s of the curve. :

Example 1. Consider the helix in the parametrization

-y(t) = (a cos t, a sin t. b t) . a' + 2 , the length function is L(t) = a + b2 t. From

Since jjy(t)Ij = this, we obtain the natural parametrization by passing from the parameter

t to the parameter s = L(t):

(s) _

s

a2+b2

s a2 +61

To every curve, we shall assign three linearly independent vectors at each point of general type. The first of these vectors is the normalized tangent vector.

S. Curves and Surfaces in Euclidean 3-Space

132

Definition 2. If y(s) is a curve in its natural parametrization. the vector t(s) := y'(s) is called the unit tangent vector to the curve.

The curvature of a curve at a point measures the angle variation of the tangent vector per length unit. More precisely: Definition 3. Let C C R3 be a curve, and let p E C be a point on the curve. The curvature K(p) of the curve C at the point p is the limit

tc(p) = lim 4

q-p pq where o is the angle between the tangent vectors at the points p and q. and pq denotes the length of the curve segment between these points.

Theorem 2. In the natural parametrization of the curve, :(s) = IIy"(s)II. Proof. Setting p = y(s) and q = y(s + h), we obtain Tiq = h in the natural parametrization of the curve, and the angled is computed using the formula for the angle in an isosceles triangle,

sin(0/2) = IIt(s+h) - t(s)II/2. Thus

0

lim q--p pq

2 sin(¢/2) = lim Il t(s + h) - t(s)II 0/2 = lim = I Iti (s)II h-.0 sin(0/2) h-0 h-0 h h

In an arbitrary parametrization of the curve, a straightforward parameter transformation yields the following formula (see Exercise 1):

K(t) = II''(t) x y(t)II Ily(t)II3

Definition 4. The principal normal vector h to a curve at a point of nonvanishing curvature is the normalized derivative of the unit tangent vector,

h(s) := 1 t'(s) = y(S) n(s)

11-y" (s)II

The vector product

b(s) := t(s) x h(s) is called the binormal vector to the curve.

Thus, at each point of the curve with non-vanishing curvature, there exist three mutually orthogonal vectors F(s), hh(s), and bb(s) all of length one, the so-called Frenet frame of the curve. Finally, we introduce a last geometric characteristic of a space curve, the torsion.

5.1. Curves in Euclidean 3-Space

133

Definition 5. The torsion of a curve C c R3 of class C3 at a point of non-vanishing curvature is the scalar product d6(s) r(s) _ -( ds

The formula for the torsion in an arbitrary parametrization is the subject of Exercise

1.

The structural equations of a curve in euclidean space are the Prenet formulas expressing the derivatives of the P enet frame through this same frame.

Theorem 3 (Fundamental Theorem of Curve Theory). Consider a curve of class C3 with nowhere vanishing curvature in the natural parametrization. Then, in the above notation, d

t(s)

0

t'(s)

d

h(s)

-ac(s)

0

r(s)

h(s)

b(s)

0

-r(s)

0

b(s)

=

0

K(s)

Let y, [0, L] R3 be two curves with coinciding curvature and torsion in their natural parametrizations. Then there exists a euclidean motion A : 1R3 - R3 such that y`(s) = A o y(s). Let ac(s) be any positive function, and

let r(s) be an arbitrary function, both defined on the interval [0, L]. Then there exists a curve whose curvature and torsion are the given functions.

Proof. By the definition of the principal normal vector, we have t'(s) = K(s) K(s). Since 11h(s)II __ 1, the derivative k(s) is orthogonal to K(s). Hence it can be represented as a linear combination of t(s) and 9(s). The computation

(dh(s), t(s)) _

= -K(s) (h(s), t(s)> - (h(s), j.f1> s

and the definition of torsion imply the second of the Frenet formulas. But then d9(s) ds

_ O(s) ds

x h(s) + t x

dh(s) ds

= _r(s) h(s).

Viewing the vectors of the Frenet frame as row vectors,

A :=

t(s) h(s) b(s)

St, where defines a (3 x 3) matrix A of functions on [0, L] satisfying the entries of the skew-symmetric matrix Q of 1-forms are the curvature and

5. Curves and Surfaces in Euclidean 3-Space

134

the torsion of the curve: Q=

0

a(s)

0

-K(S)

0

r(s)

0

-r(s)

0

ds.

Existence and uniqueness of a curve forgiven curvature and torsion functions thus follow immediately from Chapter 4, Theorem 6, since the integrability

condition. dil - Q A Q = 0, is satisfied for trivial reasons. Note that the solution of the system of linear differential equations, dA = Q A. is defined on the whole interval on which the curvature ac(s) and the torsion r(s) are given. Having first determined the matrix A(s) for the prescribed functions K(s). r(s) from the equation dA = Q A, we obtain the curve -y(s) by one more integration.

7(s) = Jt(s).ds. Remark. The plane spanned by the tangent vector f and the principal normal vector h at a given point is called the osculating plane of the curve

at that point. The binormal vector 6 is perpendicular to this plane. The third Frenet formula implies Ir(s)I = 110(s)II, and hence the absolute value Ir(p)l of the torsion of a curve at a point p can be described as the limit

IrQ)I = v m where v denotes the angle between the osculating planes to the curve at the points p and q (compare the proof of Theorem 2). Thus the absolute value of the torsion measures how much the curve "winds out" of the osculating plane.

Next we discuss some curves with special properties, and explain how the Frenet formulas can be applied to study them.

Definition 6. A curve whose tangents form a constant angle with a fixed direction in R3 is called a slope line'. Straight lines and helices are slope lines. These can be characterized by the fact that the quotient r(s)/tc(s) is constant.

Theorem 4 (Lancret, 1802). A curve of class C3 with nowhere vanishing curvature is a slope line if and only if r(s)ln(s) is constant.

Proof. If there exists a vector a E R3 such that (t(s), a) is constant, then, differentiating this equation, we obtain that the scalar product (h(s). a) 1 In German. such a curve is called Boschungslinie.

5.1. Curves in Euclidean 3-Space

135

vanishes, since K(s) # 0. Differentiating once again leads, using the Frenet formulas, to T(s) (6(s), a = c(s) .

The vector d lies in the if, 9}-plane, and (t, a) is constant. Hence (9, d) is also constant, and thus the quotient T/rc is constant. Conversely, if r/K is constant, then consider the vector T t(s) + 6(s). K

The Frenet formulas imply ds

I£

ds + ds =

T h - T h = 0,

and hence a is a constant vector. The scalar product (t(s), d) = T/rc is also constant, and the curve is thus a slope line. O Curves lying on a sphere in 3-space can be described by a similar relation between curvature and torsion.

Theorem 5. A curve of class C4 with nowhere vanishing curvature and torsion, K, T 96 0, lies on a sphere of radius R > 0 if and only if it satisfies

_I + K2

,i

2

= R2

rc2T

in the natural parametrization. Proof. Differentiating the equation II-'(S)II2 ° R2, we obtain (t(s), -Y(s)) _ 0, and hence y(s) is a linear combination of the vectors h(s) and 6(s),

y(s) = a(s) h(s) +,13(s) b(s) . Furthermore, IIy(S)112 - R2 implies a2(s) + f32(s) (t(s), y(s)) - 0 again leads to K(s) (h(s), y(s)) + 1

R2. Differentiating

0, and thus

a(s) = We differentiate the equation c(s) (h(s), y(s)) + 1 - 0 and obtain the following relation by a simple transformation:

2s) ic'(

a(3)

I£ (S)T(S)

The asserted necessary condition for a spherical curve then immediately follows: R2

=

1

a2(s) + $2(s)

=

a2(s) +

2

r

( K2(3)T(S)

/

5. Curves and Surfaces in Euclidean 3-Space

136

If, conversely, this relation between curvature and torsion holds for a C4curve, then we first differentiate it and obtain the equation

r(s)

d ds

/G(S)

_

KG(s)

!GZ(s)r(s)

0.

Now consider the vector

a(s) := 7(s) +

.6

h(s)

,z() (s)

;R-S)

(s).

Using the Frenet formulas and the preceding relation between curvature and torsion, we compute the derivative of the latter, and find that

id(s) = 0. Hence d:= a(s) is constant, and 11-y(S)

-

aI1z

=

az(s) + \az(s)(s)/z = Rz'

i. e., the curve -y(s) lies on the sphere of radius R with center d.

0

Next we turn to plane curves. Note first that these can be described as the curves with vanishing torsion. Theorem 6. A curve of class C3 with nowhere vanishing curvature, k(s) 0, lies in a plane in R3 if and only if its torsion r(s) - 0 vanishes identically.

Proof. Let a' be a vector perpendicular to the plane in iR3 containing the curve. All the tangent vectors t(s) lie in this plane; hence (t'(s), a) - 0. Since ac(s) # 0, we immediately obtain (h'(s), a") - 0 by differentiating this equation. Thus d coincides with the binormal vector 6(s). In other words, the binormal vector 6(s) is constant. Then 0 = 6'(s) = -r(s) h(s) implies that the torsion vanishes, r(s) - 0. The converse is proved analogously. 0 The curvature of a plane curve can be ascribed a sign. In fact, the principal normal vector is proportional to the vector obtained by rotating the tangent vector through the angle 7r/2 in the positive sense. The curvature of a plane curve is ascribed the positive sign if the corresponding factor is positive. Identifying R2 with the complex numbers, the rotation through 7r/2 in the positive sense corresponds to multiplication by the number i E C. Using the multiplication of complex numbers, this leads to

Definition 7. Let -y : [0, L] - C = R2 be a plane curve in its natural parametrization. The plane curvature k(s) is the function k : [0, L) -' R defined by the equation ast(s) = k(s)

i

t(s) .

5.1. Curves in Euclidean 3-Space

137

The absolute value jk(s)j of the plane curvature coincides with the curvature K(s) of the curve viewed as a space curve.

Example 2. -y(t) = (t, ±t2)

.

A closed curve i' : [0, L] - C = R2 is one that starts and ends at the same point and whose tangent vectors at this point coincide as well, i'(0) = -Y(L) and t(0) = F(L).

Theorem 7. Let

C be a closed curve. Then the integral

[0, L]

r

2 J7

k(s)ds =

2

ILL

is an integer, called the winding number of the closed curve.

Proof. Consider the map t' : [0, L]

t* (s) = exp

C defined by

[k(u)du]

Then dt'(s)/ds = i k(s) - t*(s). and hence (t(s)/t*(s))' = 0. The tangent vectors t(s) are thus described by the formula

F(s) = C exp

Lk(u)duj fL

for a certain constant C. Since t(0) = t(L), the number J k(s)ds is an integral multiple of 27r.

o

0

We conclude the section on curves with the discussion of the Fenchel inequality, which claims that the total curvature of a closed space curve is

5. Curves and Surfaces in Euclidean 3-Space

138

bounded from below by 2;r. We start by considering plane curves, and then generalize the result to space curves. First we need an auxiliary observation. Lemma 1. Let cp : [a, b] lR be a real function of class C', and suppose that the function f (t) := ew<
(1) f If(u)Idµ 2: ir; b a

(2) in this estimate, equality holds if and only if the derivative cp does not change sign, and, moreover, IV(b) - p(a)I = r.

Proof. From j (p) = i

we obtain

f If (j)I dµ ? I f b

b

(p)dui = Iw(b) - w(a)I.

Since f (a) = 1 and f (b) = -1, there exist integers k, I such that W(a) = 2kir and yp(b) = 21;r + tr. This implies

f bIf(i)Idy > irI1+2(k-l)I > a. a

0

The case of equality follows immediately.

Theorem 8. Let -y

:

[0, L] - R2 = C be a closed plane curve. The total

curvature of the curve is at least 27r:

jL jIk(s)Ids =

> 2ir.

Equality in this estimate occurs if and only if the closed curve -y bounds a convex region.

Proof. To every point of the curve we assign the tangent vector to the curve.

This determines a map from the interval to the unit circle S' satisfying F(0) = F(L). The image set F([0, L]) C S' is a compact and connected subset

of S', hence an arc. This arc contains at least one pair ±zo of opposite points. Otherwise, there would exist a number zo E S' such that neither +za nor -zo would belong to the arc. But then o F(s) would have an imaginary part which would have constant sign, and from

J:Im Ga F(s)1

ds = Im -

\ Za

f L F(s)ds) = Im

\ Z;

(y(L)

- y(0))) = 0

we would obtain a contradiction. Without loss of generality, we may assume

zo = I and r(O) = F(L) = 1. Moreover, there exists a parameter value

139

5.1. Curves in Euclidean 3-Space Si E [0, L] such that r(s1) _ -1. Applying the lemma above, we obtain

f

0sl

81

ik(s)Ids = f IZ'(µ)Idi

>- 7r,

0

f

L

Ik(s)Ids =

f

L 71,

81

81

L

Ik(s)Ids >_ 2a. The inequality for plane curves is thus and, altogether, J proved. In the case of equality, the lemma implies that the plane curvature k(s) of the curve does not change sign on the interval [0, s1], and /081

k(s)dsI

= it = 14LL k(s)dsl

.

Now we exclude the case that k(s) has different signs on the intervals [0, s1] and [s1, L]. If this were the case, then k(0) = k(si) = k(L) = 0,

and k(s) would be positive on the interval [0, s1] and negative on [s1, L]. But then the inequality

0<

f

k(µ)dp < 7r

would hold for all parameters s E [0, L], and the tangent vector t(s) _ y

exp

k(p)dµ] would stay in the upper half-plane (z E C : Im

hence we derive a contradiction to L

t(s)ds = -t(L) - ^t(O) = 0. The plane curvature k(s) of the curve thus has constant sign, k(s) > 0. In this case, the whole curve completely lies on one side of the tangent to the curve at an arbitrary point. In fact, without loss of generality, we can choose the point as that corresponding to the parameter value s = 0. Consider the function /3(s) measuring the height of the curve point -y(s) over the tangent through -t(0),

Q(s) := Re (('Y(s) - Y(0)) (i t(0)) The derivative of Q(s) is easily computed:

13'(s) = Re

00) exp I i f e k(u)du] Ti t(0))) L

= sin

(j8

k(u)du)

o

The curvature is non-negative, and thus /3(s) is negative for parameter values

close to s = 0. If /3(s) had any negative value in the interval [0, L], then there would exist a value snin such that /3(smin) < 0 and Q'(smin) = 0.

5. Curves and Surfaces in Euclidean 3-Space

140

/L Since

J0

k(u) du = 21r, this value s is determined by the equation 5min

k(u) du = 7r,

and, for s E [0, smin], the derivative of the function 0(s) would be negative. Therefore, $(s) would not be decreasing in [0, smi,,], contradicting /3(smio) <

0. Summarizing, we obtain that the height function is non-negative, and thus the curve lies completely on one side of each of its tangents. Consider now the half-planes determined by all the tangents. Their intersection is a convex domain whose boundary coincides with the curve ry(s).

0

We generalize the inequality of the preceding theorem to space curves. To do so, we need a preparation.

Lemma 2. If D C S2 is a closed curve in the sphere of radius one whose length does not exceed 2a, then V is contained in a hemisphere. Proof. Choose two points P and Q on D which divide the curve into two segments of equal length, V = D1 UD2. Let the north pole N = (0, 0, 1) lie on the shorter segment of the great circle through P and Q on the sphere. Denote by Di the curve segment obtained from Dl by rotation around the z-axis through the angle a. The union Dl U Di is a closed curve on the sphere whose length coincides with that of the original curve D. If the curve segment Dl intersects the equator S' = {(x, y, 0) : x2+y2 = 1} of S2, then the curve Dl U DI contains opposite points on the sphere. Hence its length has to be at least 27r. By assumption this length cannot exceed 27r. Thus 7)i does not intersect the equator S' at all, if the length of the curve D is smaller than 27r. Otherwise, D, or V2 is a semi-circle on the equator, respectively. In both cases, the curve D is contained in the upper hemisphere. 0 Q

5.2. The Structural Equations of a Surface

141

Theorem 9 (Fenchel Inequality). The total curvature of a closed curve C C R3 is at least 27r, Jc

K(s) ds > 27r.

Equality holds if and only if C is a plane curve bounding a convex domain.

Proof. Let a non-plane curve be given. Consider its tangent map t : [0, L] -+ S2 as a "curve" in the sphere S2. The length of the latter coincides with the total curvature of the original curve, fL rL L(t) = I

Jo

0

Note first that the image of t cannot be contained completely in a hemisphere

of S2. If this were the case, then there would be a vector a' E S2 such that (a, F(s)) _> 0 for all values of the parameter. But the integral of t(s) vanishes, since we started from a closed curve. Hence (d, F(s)) - 0, and the curve C would be a plane curve. The assertions of the theorem now follow from the previous lemma together with Theorem 8.

5.2. The Structural Equations of a Surface A surface is a two-dimensional submanifold of 1R3. As in the case of curves, we want to allow for self-intersections and will also consider parametrized

surfaces. This is meant to denote a map F : U --+ R3 defined on an open subset U of R2 (or, more generally, on a two-dimensional manifold U) with

the property that the differential D(F) has rank two at each point. As in the case of a two-dimensional submanifold A12 C l3, we denote by M : M2 - R3 the inclusion into 3-space and view this map as a vector-valued function. Locally, we choose an orthonormal tangent frame el, e2 to the surface and denote by e3 := el x e2 the normal vector to the surface. We sometimes consider-without introducing a new notation-el, e2 and e3 also as vector-valued functions on the manifold 1112. Denote by al and o2 the frame of 1-forms dual to {el, e2}. The differential dM of the identity map on the surface is the identity map on each tangent space to the surface. Hence the following equation holds:

dM =01 - el + expressing nothing but the decomposition of a tangent vector with respect to the basis e1, e2. The exterior product of A o2 is a 2-form on the surface which is independent of the choice of the orthonormal frame el, e2 for a fixed orientation and coincides with the volume form,

012 = of A a2.

5. Curves and Surfaces in Euclidean 3-Space

142

The scalar products (ei, ej) of the vector-valued functions ei are constant functions on M2. By differentiation, we obtain

(dei, ej) + (ei, dej) = 0 and denote the 1-forms by wij := (dei, ej). Now combine the differential forms into an antisymmetric (3 x 3) matrix Cl:

=

0

w12

W13

w21

0

W23

W31

W32

0

Using the forms just introduced, the differentials of the vector-valued functions ei can be represented as 3

de. = E wij ej j=1

Altogether, we constructed five differential forms of degree one, the forms of and wij. These are, however, not independent; the so-called structural equations of a surface express the relations among them. The fact that the form w12 is completely determined by 01 and 02 will turn out to be of special importance. Later on, this will imply the fact that the Gaussian curvature belongs to the inner geometry of a surface (Theorema Egregium, Gauss, 1827).

Theorem 10 (Structural Equations of a Surface). (1) d01 = W12 A C2, do2 = w21 A o1 . The form W12 is completely determined by the forms of and o2

(2) a1 A w13 + 02 A w23 = 0 (3) dC = n A Cl

.

.

.

Proof. Differentiate the function M twice,

0 = ddM = d(o 1 - el + 02 e2) = (dal - w12 A 02) el + (do2 - w21 A 01) e2 - (a A w13 + 02 A w23)

e3 .

This yields the equations stated in (1) and (2). The identity for Cl follows using 0 = ddei = Ek (dwik - Ej wij A wjk) ek. The form w12 can be computed from al and 02 as follows. Represent the differentials d01 = A 01 A o2i

d02 = B 01 A o2

as multiples of the 2-form of A 02. The first structural equation implies w12 =

Aa1+Bo2.

5.2. The Structural Equations of a Surface

143

Let a parametrization F : U - M2 C R3 of a surface by a a domain U C R2 be given; then the 1-forms ai and wij can be pulled back to U. The

resulting forms will be denoted by a, := F'(ai) and wij := F'(wij). Then the structural equations of the surface hold as well: 2

do,; _

wij A aj*,

al A W13 + a2 n w23 = 0,

dtl' = St' n it

.

j=1

Now we prove a first formulation of the fundamental theorem of surface theory. This allows to determine a surface locally in 3-space from the system of associated 1-forms satisfying the structural equations relating them.

Theorem 11 (Fundamental Theorem of Surface Theory-First Formulation). On a simply connected, open subset U of R2 let four differential forms 01, a2, 013, 023 of degree one be given. Suppose that the forms 01 and 02 are linearly independent at each point. Define the 1 -form 012 by the equations

d'1 = 012/02, d02 = -012A01. Extending them antisymmetrically, wji := -wij, assume that the given system satisfies the structural equations U1 A 01$ + a2 A 023 = 0,

dfl = NA N.

Then there exist a parametrized surface F : U -' R3 and an orthonormal frame of tangent vector fields such that the induced forms a, and wij coincide with the original forms aFi and wij, respectively. The surface together with its orthonormal frame is uniquely determined up to a euclidean motion of R3.

Proof. The (3 x 3) matrix S2 is skew-symmetric and satisfies the integrability condition dSl = SE A li. For a given initial condition Ap E SO(3, R), there exists a matrix q of functions such that St A = dA by §4.3, Theorem 6. The

rows of this matrix are orthogonal and have length one. Thus they define three vector-valued functions e1, e2, e3 : U - R3, and 3

dei = j=1

Consider the 1-form

0 := 01.91 + 02.92 Computing the differential of 0 yields

dO =

0.

By Poincares' lemma (§3.6, Theorem 22) there exists a map F : U - R3 such that dF = ¢, = Q1 e1 +02 e2i and hence we obtain the parametrized surface together with the orthonormal frame. Twice in this proof we integrated a

5. Curves and Surfaces in Euclidean 3-Space

144

differential equation, and thus there remains a certain degree of freedom with respect to the initial condition. First, the solution q of the differential equation Il A = dA depends on the prescribed orthogonal matrix Ao E SO(3, R). In the second step, the function F : U -' R3 is determined by the differential equation dF = ¢ only up to a constant vector a E 1R3. Summarizing, we see that the surface is, for prescribed forms of and wtj, determined up to a euclidean motion. Example 3. We discuss the case of a surface of revolution in R3. Consider a plane curve in the (x, z)-plane with component functions r(s) > 0 and z(s). The surface F arises by letting this generating curve revolve around the z-axis, and thus it has the following parametrization defined on U = 1[2 x (0, 27r):

F(s,cp) = (r(s)sincp, r(s)cosco, z(s)).

For simplicity, we suppose that the generating curve of the surface F is parametrized by arc-length, i.e., at each point the following condition is satisfied:

= 1.

(r')2 + (Z,)2

(*)

(Compare Exercise 15 for the general case.) First we compute two tangent vectors to the surface,

a = (r sin co, r cos co, z'), as

a

= (r cos V, -r sin tp, 0),

a

s

as well as their scalar products, a

s

as' as

(r,)2

=

+

(z,)2

= 1,

as 5 = 0, ,

a s--

app

asp

r2

Normalizing the second tangent vector correspondingly, we obtain the orthonormal frame el

_ as, a

e2

_ 1a

r Op

and the basis dual to it,

01 = ds,

02 = r dip.

The unit normal vector to the surface F is the vector product of el and e2,

e3 := el x e2 =

z'sin,p z'cos

-r'

We compute the differential of el = (r'sin gyp, r' cos cp, z') componentwise,

del = (r" sin cp ds + r' cos cp dcp, r" cos pds - r'sin pdcp, z"ds),

5.2. The Structural Equations of a Surface

and check that del = r' dcp e2

145

+r zI

ds e3 .

In fact, inserting e2 and e3 into this linear combination, we obtain r' dcp e2 + ; ds e3 z

= (r' cos cp dcp, -r'sin cp dcp, 0) +

z, (z'sin cp dcp, z' cos cp ds, -r' ds)

= (r" sin cpds + r' cos cp dcp, r" cos cp ds - r'sin cp dcp, -r" r'/z' ds) .

But, since differenting the relation (*) implies rr' + z'z" = 0, this is equal to del. Similarly, one proves that

de2 = -r'4 el - z' dcp e3,

de3 =

-r z

ds el + z' dcp e2,

and, therefore, we computed the matrix f of 1-forms w, completely: 0 -w12 -w13

W12

w13

0

w23

-w23

0

0

=

-r dcp

: ds

Z' ds

r dcp 0

-z'dp

z' dcp

0

In addition, we show in this example how to compute the form w12 directly

from al and a2. Starting from dal = 0 and dal = r'-dsAdcp = r/r-al Aa2, we obtain the formula

W12 = r -a2 = r - dtp using the method described in the proof of Theorem 10. Computing w12 directly from the definition turns out to be even easier:

r'

r' dcp w12 = (del, e2) = (r'dcp e2 + z' ds e3, e2) Example 4. Let h : U R be a real function in two variables on the open set U C R2, and denote by F its graph, F(x,y) = (x, y, h(x,y)).

We indicate partial derivatives by lower indices. Starting from the two tangent vectors a ax = (1, 0, h=) and ay = (0, 1, hy), we determine an orthonormal frame with the same orientation,

e, =

a

-+h2 x+ 1

hZhy

e2 _

h2

y 1+ by

1 + h2 + h2

l

a- a

h,hy ay

ax lJ

5. Curves and Surfaces in Euclidean 3-Space

146

as well as the 1-forms dual to these vectors,

of =

1+h2dx+

VT T h2 + h2

hxhy

1+h2 dy'

Q2

l+h2

=

dy.

The corresponding volume element is

1+hz+hndxndy.

of Aa2 =

The computation of the whole matrix fl is more extensive. Because of its geometric relevance, we give the form w12 for later use: W12 = (del, e2) =

hu dl

(1+h2

1+h2+h2

In the discussion of curvature properties of surfaces, we will see that particularly interesting graphs arise from holomorphic functions. Therefore, these are discussed next.

Example 5. Let f : U

C be a holomorphic function. The modular surface of f is the graph of the modulus function h = if I = (fl) 112 of f. The relations between the derivatives with respect to the complex parameter z and the (real) partial derivatives are expressed in the following formulas:

8

8z -

1

a

2.) ,

8x - z by

8 8z

-

1

2

8 8 ex + i FY)

Hence it is possible to express hx and by by the z-derivatives of f and f. The latter are denoted by '. It is easy to see that 8h _ f'f 8h _ ff' 8z ]F h-' 8z 2h Thus the derivatives we wanted to compute are

hz = 2h(f'f +ff) and by = Rewriting this yields, e. g. ,

1+h2+h2 = 1+1992, and inserting these expressions into the formula of the previous example we immediately obtain an explicit expression for the corresponding form w12 depending on the complex derivatives of the function f. Indicating the lines of constant modulus as well as those of constant argument on its modular surface leads to a clear picture of the behavior of the function in the complex domain. As an example, we consider the function f (z) = 1/ sin(z) outside its poles, shaded according to the argument and showing the level lines of the modulus:

5.3. The First and Second Fundamental Forms of a Surface

147

Example 6. Decompose a holomorphic function f : U - C into its real and imaginary parts, f = u + iv. Taking in Example 4 for h the functions h = u and h = v, respectively, we obtain the graphs of the real and the imaginary part of f. Starting from the Cauchy-Riemann equations,

ux = Vy,

uy = -vx ,

and first using the formula for 8182 derived in the preceding example, f _ ux - iuy = vy + ivx, we obtain

1+uz+uy = 1+vv+v22 = l+If'I2. At this point we omit the explicit formula for w12.

5.3. The First and Second Fundamental Forms of a Surface In classical differential geometry, it is common to describe a surface by two

square matrices, the so-called first and second fundamental forms. The first of these is already familiar from Chapter 3, since the first fundamental form is just the Riemannian metric. The second fundamental form depends on the particular realization of the surface in euclidean space. To be able to define the first fundamental form in the language of differential forms,

5. Curves and Surfaces in Euclidean 3-Space

148

we introduce the symmetric product of 1-forms, in addition to the exteror product we already know.

Definition 8. Let M' be a manifold, and let a, µ be two 1-forms on Mn. Define a symmetric bilinear form a 0 p : TM" x TM" IR by a 0 µ(V, W) := 2 (a(V)µ(W) + a(W)µ(V)) . In the literature, it is common to omit the symbol 0. The symmetric square of a 1-form or is often denoted by a2.

The definition immediately implies that, on a surface with orthonormal frame el, e2 and corresponding dual basis of 1-forms al, a2, the following equation holds:

a: O al(ej, ee) = ai(ei)a1(et) = bijb k For the basis e1, e2 the bilinear form

I := al 0ol+a2Oa2 is thus precisely the usual euclidean scalar product. It is called the first fundamental form of the surface; it coincides with the Riemannian metric defined in Chapter 3. Geometric quantities defined for the surface depending only on the first fundamental form are called quantities belonging to the inner geometry.

Example 7. For the surface of revolution from Example 3, we had al = ds and a2 = r dip; thus the first fundamental form is I = ds O ds + r2dcp O dcp. This is also written as I = ds2 + r2dc 2.

Example 8. The first fundamental form of a graph (Example 4) is I = (1 + h2) dx2 + 2 hhxhy dx dy + (1 + h2) dy2 .

Example 9. On the upper half-plane x2 = {(x,y) E R2 : y > 0}, we consider the 1-forms

al = dxy-, a2 =dyy-. They determine the Riemannian metric dx2 + dye y2

The half-plane N2 endowed with this metric is called the hyperbolic plane. The form w12 depending only on al and a2 is equal to al.

5.3. The First and Second Fundamental Forms of a Surface

149

If V and W are two tangent vector fields to the surface M2, then, viewing W as a vector-valued function and differentiating it as such with respect to the vector field V, we denote the resulting vector-valued function by V(W). In general, V(W) is not tangent to the surface. However, Theorem 12. Let V and W be two tangent vector fields on a surface. Then the difference V(W) - W(V) is also tangent to the surface, and it coincides with the commutator of the vector fields,

V(W) - W(V) = [V, W]

.

Proof. In the proof we make use of the structural equations of the surface. V and W are tangent vector fields, and hence the scalar products (V, e3) _ (W, e3) = 0 vanish identically. This implies (V(W) - W(V), e3) _ - (W, de3(V)) + (V, de3(W)) - (W, w31(V)el +w32(V)e2) + (V, w31(W)el +w32(V)e2) al A w31 (V, W) + a2 A W32 (V, W)

Because of the structural equation a1 A w31 + a2 A w32 = 0, the difference V(W) - W(V) is a tangent vector field on the surface. Moreover, from §3.9 and the structural equation, we conclude for i = 1, 2 that

a;(V(W) - ),V(V)) = (V(W) - W(V), e;) = V((W,e1)) - (W,de1(V)) - W((V,e1)) + (V,de;(W)) 2

= V((W,e;)) - W((V,e:)) +1: aj Awii(V,W) i=1

= V(a;(W)) - W(a2(V)) -dai(V,W) = a+([v,WI). Thus V(W) - W(V) coincides with the commutator [V, W].

0

We decompose the vector-valued function V(W) into its tangent and its normal part. The tangent part is denoted by V W, whereas the normal part will give rise to the so-called second fundamental form II(V, W) of the surface. Definition 9. Let V and W be tangent vector fields on the surface M2 C R3. The tangent part of V(W),

V W := V(W) - (V(W), e3) e3 = V(W) + (W, de3(V)) e3, is called the covariant derivative of the vector field W with respect to the vector field V. The normal part of the vector-valued function V(W) will be denoted by II(V, W), II(V, W) := (W, de3(V)) :

5. Curves and Surfaces in Euclidean 3-Space

150

it is called the second fundamental form of the surface.

Remark. The second fundamental form of a surface depends on the orientation of M2. Changing it reverses the normal vector e3, and hence II changes sign, too.

First we collect the properties of the covariant derivative V.

Theorem 13. (1) Vv(W1 + W2) = Vv(W1) + Vv(W2);

(2) Vv,+y,W = Vy,W+Vv,W; (3) If f : M2

R is a smooth function, then

Vv(f W) = df (V) W + f VyW and V1.yW = f VyW; (4) V (W1, W2) = (VVW1, W2) + (W1, VvW2);

(5) VyW - VwV = [V, W). Proof. The additivity of VvW is a straightforward consequence of the definition. The last formula follows from the previous theorem. All the remaining assertions are easily checked, e. g.,

Vv(f - W) = V(f W)+(f - W,

VyW.

Representing the vector field W in an orthonormal frame e1, e2 on the sur-

face, W = W' e1 + W2 e2, and taking into account that VVeI = del(V) + (el,de3(V))

e3

= w12(V) . e2 + w13(V) e3 + wsi(V) e3 = w12(V) e2,

we obtain the important formula

VvW = (dW' (V) + w21(V) W2) - el + (dW2(V) + w12(V) W') e2 . The form w12 depends only on a,, a2i and hence the covariant derivative VvW also depends exclusively on quantities belonging to the inner geometry of the surface. The second fundamental form lI(V, W) is a symmetric bilinear form on the tangent bundle of the surface. Theorem 14. Let V and W1, W2 be tangent vector fields on a surface, and let f1, f2 be smooth functions. Then (1) the second fundamental form is symmetric, II(V, W1) = II(Wi, V);

(2) II(V, f1 Wi + f2 W2) = f1 II(V, Wi) + f2 II(V, W2).

5.3. The First and Second Fundamental Forms of a Surface

151

Proof. We prove the symmetry of the second fundamental form; the remaining property immediately follows from the definition. With V and W, the commutator [V, W] = V(W) - W(V) is a tangent vector field as well. Thus

(W(V), e3) = - (V(W), e3) + (V(W) - W(V), e3)

(V, de3(W))

_ -(V(W), e3) = (W, de3(V)), i.e. , II(V, W) is symmetric.

Because of the structural equation de3 = W31 el + W32 e2, the second fundamental form can be written as a symmetric product of the following 1-forms: II = W31 O Q1 + W32 O v2.

Hence, in the orthonormal frame el, e2, the second fundamental form is represented as the symmetric matrix II(e1, el) II = I II(e1,e2)

II(el, e2)1

w31(el)

(31(e2)

II(e2,e2)

w32(el)

w32(e2)]

If, on the other hand, F : U - M2 C R3 is a parametrization of the surface, and we choose the normal vector

_

-

F X OF

1

(OF

Ig-

OF

8F

X Oy2)

Oy1 57 then the second fundamental form is given by the symmetric (2 x 2) matrix e3

II e-Fr i

b_ with coefficients

_

82F

,

e3)

11

b11

b12

b21

b22,

1

82F

OF I

OF

W2/

So for each surface M2 C R3, there exist two symmetric bilinear forms, I and II, of which the first is positive definite at each point. These two fundamental forms are not independent; the structural equations of the surface will lead

to a pair of differential equations relating them to each other. In order to formulate these equations, we introduce the curvature tensor of the surface. This is a transformation R : TM2 x TM2 x TM2 - TM2 associating with every triple of vectors a tangent vector.

Definition 10. Let U, V, W be tangent vector fields on the surface. The curvature tensor R is defined by the following formula:

R(U, V)W := VuvyW - VvvuW - vu,vlW .

5. Curves and Surfaces in Euclidean 3-Space

152

The following relations for the curvature tensor are a formal consequence of the properties of the covariant derivative VvW for vector fields stated in Theorem 13. We leave their verification to the reader. Theorem 15. Let U, V, W be tangent vector fields, and let f be a junction. Then

R(U,V)(f'W) = f.R(U.V)W.

(1)

(2) R(U, V)W = -R(V,U)W, (3) I(R(U, V)W1, W2) = -I(R(U, V)W2, W1), (4) I(R(U, V)W1, W2) = I(R(W1, W2)U, V),

(5) 1(U.V)W+7Z(V,W)U+7Z(W,U)V = 0. Recalling that the covariant derivative VvW of two tangent vectors only depends on the forms o1, o2 belonging to the inner geometry, the same also

holds for the curvature tensor. We prove a local formula for R(U, V)W specifying this fact.

Theorem 16. Let e1, e2 be an orthonormal frame on M2. Then

R(U,V)W = dw12(U, V) (°t(W) e2 - o2(W) e1)

.

Proof. Because of the symmetry properties of the curvature tensor stated in the preceding theorem, it suffices to prove this formula for the basis fields

U = Cl. V = e2, W = e1. In this case, 1Z(e1,e2)e1 = Vei(Ve2e1) - Ve2(Veiel) - V[eI.e2]e1 V'. (w12(e2) . e2) - Ve2(u12(el) ' e2) - W12([el e2])

- e2

(el(w12(e2)) - e2(w12(el)) - W12([el,e2]))'e2 + (wt2(e2)W21(e1) - w12(et)w21(e2))'el

dw12(el,e2)'e2,

and this last expression coincides with the right-hand side of the equation to be proved. The first fundamental form is positive definite. Hence the symmetric second fundamental form can be represented by a symmetric endomorphism S

T A,12 - TAP defined by

II(V, W) = I(V, S(W)) , the so-called Weingarten map of the surface (see §5.4). The equation II(V, W)

(V. de3(W)) implies that S can be rewritten as

S = de3 = W31 ' el + w32 e2 .

5.3. The First and Second Fundamental Forms of a Surface

153

The covariant derivative of an arbitrary endomorphism E : TM2 - TM2 TM2 with values in of the tangent bundle is a 2-form VE : TM2 x TM2 TM2. This is defined by the equation

VE(V, W) :_ Vv(E(W)) - Vw(E(V)) - E([V. W]). Theorem 17 (Gauss and Codazzi-Mainardi Equations). (Codazzi-Mainardi equation); (1) VS = 0 (2) IZ(U, V)W = II(V,

(Gauss equation).

Proof. Using the formula for the exterior derivative of a 1-form, the equation dS = dde3 = 0 first implies

0 = V(S(W)) - W(S(V)) - S([V, W]). Looking at the tangent parts and taking into account that S([V, W]) is already tangent, we obtain

0 = v (S(W)) - Vw(S(V)) - S([V, W]). Hence VS = 0 is proved. Recalling that the second fundamental form is II(V. W) = (V, de3(W)), we obtain S(U) = de3(U). Using now de3 = w31 el +W32 - e2 and then applying the preceding theorem proves the Gauss equation: II(V, W) S(U) - II(U, W) S(V) = w13 A w32(U, V) (o l (W) e2 -

el )

= dw12(U,V) (al (W) e2 - o2 (W) e1)

= R(U, V)W.

0

Next we will prove that prescribing a positive definite first fundamental form as well as a second fundamental form uniquely determines a surface up to a euclidean motion in R3, provided that both fundamental forms satisfy the Gauss and the Codazzi-Mainardi equations.

Theorem 18 (Fundamental Theorem of Surface Theory-Second Formulation). Let two symmetric bilinear forms 1, II : TU x TU -+ R be given on a simply connected, open subset U of R2, and suppose that the first, 1, is positive definite at each point. Starting from 1, the equation 21(VUV, W) = U(1(V, W)) + V(1(U, W)) - W(I(U, V)) + 1([U, V]. W) +1(V, [W, 141) - I(U, [V, W]) defines a covariant derivative V for vector fields with respect to vector fields, and the curvature transformation is determined by

R(U,V)W := VuvvW - VvVuW - v[U,V]W. If the symmetric endomorphism S : TU I(U, S(V)) satisfies

TU defined from II by II(U, V) _

5. Curves and Surfaces in Euclidean 3-Space

154

(1) V S = 0,

(2) R(U, V)W = II(V, W) -3(U) - II(U, W) S(V), then there exists a parametrized surface F : U -+ R3 such that the induced first and second fundamental forms of this surface coincide with I and II,

1 = F'(I), II = F'(II) . This parametrized surface is uniquely determined up to a euclidean motion of 1113.

Proof. The proof is based on a straightforward application of the fundamental theorem of surface theory in its first formulation (Theorem 11, §5.2).

To apply this, we choose a frame {ei, e2} of vector fields on the open set U C JR2 which is orthonormal with respect to the bilinear form I, and denote by 01, 02 the corresponding dual frame of 1-forms. Define additional 1-forms by the equations w12((V) := Wvei, e2), 013(V)

11(V, e1) = 1(S-(V), el),

i23(V) := R(V, e2) = 1(3(V), e2)This leads to a system {ai, Q2, "12, W13, X23} of 1-forms, which is extended

by requiring antisymmetry, wji := -Oil. The equations we supposed to hold imply that the integrability conditions of the fundamental theorem of surface theory in its first formulation are satisfied. To see this, note first that the equation defining the covariant derivative V immediately yields the formulas

U(I(V, W)) = I(VuV, W) +I(V, VuW), Vuv - Vvu = [u, V]

.

From these, we obtain

doi(U,V) = U(o1(V))

- V(aF1(U)) -aFi([U,v1)

= U(I(V,e1)) -V((U,e1)) -I([U,V],ei) = I(VuV - VvU - (U, V), ei)+1(V, Duel) - I(U, Vve1) = w12(U) . a2(V) - w12(V) U2(U) = 012 AQ2(U,V) Hence dv1 = 012 A o2, and similarly one verifies that dv2 = 021 A U1. In the next step, we show that v1 A 013 + Q2 A 023 = 0. This follows from

01 A 013(U, V) + Q2 A O23(U, V) = II(V,U) - II(U, V) = 0, since the second bilinear form is symmetric. For the computations to follow,

note that I(ei,ei) e 1 implies I(Vu'el,'e1) = 0, and hence Duel is parallel to e2. Thus for arbitrary vector fields U, V we always have

IMue1, Ove2) = 0.

5.4. Gaussian and Mean Curvature

155

Using this equation and the Gauss equation, we compute the form dw12:

dw12lu,V) = U(i(vvel,e2)) - V((VUel,e2)) - I(V[u,v]el,e2) = I(R(U, V)el, e2) = II(V, el) II(U, E2) - II(U, E1) . II(V, e2) = w13Aw32(U,V)

From this equation we get the structural equation to be satisfied, d012 = w13 A 032. The relation for d013 is derived in a similar way using the Codazzi-Mainardi equation:

d0134 V) = U(I(S(V),el)) -V((S(U),el)) -1(3([U,V]),El) = I(V S(u, V), el) + I(S(V), duel) - I(S(U), vvel ) = I(S(V), F2) I(e2, 7ue1) - I(S(U), e2) I(e2, vvel)

= 012A023(U,V) Thus d013 = 012 A 023, and an analogous computation yields d O23 = Together, the system of forms {Q1, a2, w12, 013, w23} satisfies the integrability conditions of Theorem 11, §5.2, and hence the existence of w21 A W13.

1R3 such that F'(I) = I and F'(II) = II a parametrized surface F : U is proved. The uniqueness statement is a consequence of the corresponding uniqueness result of that theorem. 0

5.4. Gaussian and Mean Curvature Consider a surface M2 C R3 and its first fundamental form, the induced Riemannian metric. Choosing a local orthonormal frame {el, e2} with dual frame al, a2, we know that the 1-form w12 = (del, e2) = (Vel, e2) is completely determined by the forms al, 02. Since its exterior differential dw12 is a 2-form, there exists a function G on the surface such that d012 =

The function G is independent of the choice of the local frame of vector fields. Every other frame {ei, e2} can be represented as

ei =

e2,

e2 = -µ sinap el +coscp e2, where µ = ±1 is constant, and V is a function on that part of M2 where both orthonormal frames are defined. The dual frames are thus related by

d = µ cos 92

of + sin cp o,2,

a2 = -µ sin p a2 + cos

a2,

and we obtain ai A o2 = µ of A a2. On the other hand, the form w12 is equal to w12 = (dei, e2) = µ w12 + dip, leading to the formula &.012 = µ &012This observation shows that the function G is uniquely defined.

5. Curves and Surfaces in Euclidean 3-Space

156

Definition 11. The Gaussian curvature G : M2 -* R is the function defined by the equation

dw12 = -

not.

Remark. The Gaussian curvature depends exclusively on the first fundamental form of the surface. If this is given, we can choose an orthonormal frame of 1-forms al, a2. From the equations

dal = A- al na2 and da2 = B - al n ag, we determine two functions A, B and then introduce the 1-form

wit = Finally, the defining equation for the Gaussian curvature is dw12 = This way, the Gaussian curvature can be computed in practice.

The formula for the curvature tensor of the surface stated earlier can be simplified by means of the Gaussian curvature:

R(U, V)W = G - (a2(W) el - al (W) e2) dM2(U, V), and, conversely, G can be expressed by the curvature tensor.

Theorem 19. The Gaussian curvature of a surface is G = (R(e1, e2)e2, el) ,

where 1Z is the curvature tensor, and lei, e2} is an orthonormal frame of vector fields.

Now we turn to the invariants of the second fundamental form, which we view, by means of the Riemannian metric, as a symmetric endomorphism

S : TM2 - TM2, the Weingarten map of the surface introduced in the previous section. Its normalized trace is the mean curvature of the surface.

Definition 12. The mean curvature H of a surface is the function

H := tr (S)12. The mean curvature of a surface can also be determined as the divergence (in R3) of the normal vector field. Theorem 20. Let M2 be a surface, and let N be a vector field of length one defined on an open neighborhood of M2 in R3. Assume that the restriction NJnr2 of N to M2 is perpendicular to the surface. Then

H=

1 div(N),

5.4. Gaussian and Mean Curvature

157

where the divergence is taken with respect to the euclidean metric on

R3.

Proof. From (N, dAl) = 0, we obtain

2H = (et, dN(e1)) + (e2, dN(e2)) + (N, dN(N)) The last sum, however, is independent of the particular choice of the orthonormal basis in IR3. This implies

2H =

y'-,

+(

ay j + (', Ar

= div(N) .

0

Apart from the trace, the second fundamental form S has one more invariant, the determinant. But this does not lead to a new geometric quantity, as we shall now see.

Theorem 21. The determinant of the second fundamental form is the Gaussian curvature, det(S) = G.

Proof. Let x E M2 be an arbitrary point of the surface, and choose an orthonormal basis {e1, e2} consisting of eigenvectors of the symmetric endomorphism in the tangent space TXM2. If 1C1i K2 are the eigenvalues, then,

with respect to this basis, S has the matrix rcl

0

0

K2

The assertion now follows directly from the Gauss equation, G = (1.(e1,e2)e2,e1) = (II(e2,e2)S(el), e1) - (II(el,e2)S(e2), el) = Ic1 - 1£2 = det(S).

0

Remark. The preceding theorem allows to compute the Gaussian curvature, belonging to the inner geometry of a surface, from the second fundamental form. Take an arbitrary (not necessarily orthonormal) basis v1i v2 in the tangent space TXM2 at a point of the surface, and denote the matrices of the first and second fundamental forms by g and b, respectively, 9

__

911

912

921

922

'

9ij = I(v;,ZJ),

b=

b11

b12

b21

b22J

,

bij = II(v{,vj)

The Gaussian and the mean curvature can then be computed using the formulas

and H = 2tr (b g-1)

.

det(g) Their proof is a consequence of a more general observation. A symmetric bilinear form B on a real vector space V without any additional structure has only a single invariant, the signature. If, however, a non-degenerate G

scalar product G is given in addition, then this induces an isomorphism

5. Curves and Surfaces in Euclidean 3-Space

158

G : V' - V between the vector space V and its dual space V. Thus B can be viewed as a G-symmetric endomorphism, S := B o G-t : V -. V in the vector space V, and all coefficients of its characteristic polynomial are invariants of B with respect to G. These, in turn, can be computed by means of an arbitrary basis, and the result does not depend on the particular basis. Two of these invariants of the characteristic polynomial are the trace and the determinant.

Example 10. For the hyperbolic plane 7{2 from Example 9, a straightforward computation shows that d&12 = a1 A a2. Therefore, the Gaussian

curvature is -1. A complete realization of H2 in 3-space does not exist (Hilbert, 1901). Certain open subsets of the hyperbolic plane, however, can be realized as surfaces. The related mean curvature is not uniquely determined; it depends on the particular realization as a surface in R3. One of them will be presented at the end of the discussion concerning surfaces of revolution to follow.

Example 11. For the surface of revolution from Example 3, we had

r" de3 = -z,, 4

This enables us to compute the Weingarten map explicitly:

S(a) 8s

8s

i-j

x'

8

S O'p JJ = z e2 = r 8V

Thus, in the basis 8/8s, 8/Ow, the Weingarten map is described by the matrix

S=

r"/z'

0

z'/r

0

From this, the Gaussian and the mean curvature are immediately computed:

G=-r"

r'

H=

1

(z'

2

r

r"

z')

Examples for surfaces of revolution with constant Gaussian curvature are the

cylinder (C = 0, r = 1, z = a), the sphere (G = 1, r = sins, z = cos s), or the pseudo-sphere (G = -1): The latter is an example of a (necessarily noncompact) surface of constant negative curvature, and hence a 2-dimensional model of hyperbolic geometry. It is defined as the surface obtained as the surface of revolution generated by the tractrix with the parametrization

(e_',

ja

1 - e-2tdt I = (e-', -VI - e-2' + arctanh( 1 - e-28))

.

5.4. Gaussian and Mean Curvature

159

Example 12. The Gaussian curvature of the graph of the function h(x, y) (Example 4) is preferably computed using the differential of the form w12 The resulting expression is h== hyy - hiy (1 +h2 To compute the mean curvature, it is convenient to describe the graph

the zero set of the function g : U -+ R, g(x, y, z) := h(x, y) - z. From Theorem 5, §3.2, we then know that the normalized gradient vector field, grad(g)

1

IIgrad(g)f

J+ h.2 + h2

(ham, hy, -1)1

Y

is perpendicular to the surface at each point. Hence the mean curvature can be computed as the divergence of this vector field (Theorem 20), and we obtain

H =

(1 + hy)hy= - 2h,hyhy + (1 + h2)hyy 2(1 + hZ + h2)3/2

Example 13. In order to apply this formula to the computation of the Gaussian curvature for a modular surface (Example 5), we still have to determine the second derivatives of h = If 1. From h_ 8h 8h _= Z (8h = and 8z ez + e we first conclude formally that 82h _ a2h 82h h= +2 8z2

_ huy

.92h

8z2

+2

82h _ 82h

8- 822'

- 8h) 8z J

8z8z + 0z2 ' h,,,

=

Z

102h

- 82h

8z2

8z2

5. Curves and Surfaces in Euclidean 3-Space

160

After expanding, the numerator of the Gaussian curvature turns out to be h.= hba

z - h :y

_

2 0h (92h 4 (idzoz) - a2h - a52 022

.

We compute the derivatives: a2h

fll.f

(f1f)2

f fN

O2h

92h

(fp)2

4 h3 ' azz 4 h3 ' az2 2h 2h azaz from which we deduce the formula for the Gaussian curvature:

f,z

G = (I+lf,12)2 Ref

f"-1

f, f,

4h

.

In the example discussed before, f (z) = 1/ sin z, the Gaussian curvature has the form 2

G =

Cost z

+

12 [Re

1

sill z

+ cos z

sine z

cost z - 1J 1+ cu52 z J 1212

5.4. Gaussian and Mean Curvature

161

Of particular interest is the curve on the surface separating the points of positive Gaussian curvature from those of negative Gaussian curvature. Here, it is implicitly described by the equation Re

Cost

z

=1

' 1 + cost z and is indicated as a dashed line in the figure on the previous page showing

the modular surface shaded according to the Gaussian curvature. Example 14. We now continue the discussion of the surfaces defined by the real and the imaginary part of a holomorphic function, f = it + iv : U C (Example 6). There we already derived from the Cauchy-Riemann equations the relation

l+uZ+uy = l+vi+vb = l+If'I2. A straightforward calculation yields the second derivative of f :

f" = uz2 - iu"V = v=y + iv==. The formula for the Gaussian curvature of a graph thus leads to the remarkable fact that the surfaces defined by the real and the imaginary parts have

5. Curves and Surfaces in Euclidean 3-Space

162

the same Gaussian curvature, fui2 G = (1 -I + I f,I2)2

The picture on the previous page illustrates this fact for the function f (z) _ z2, again shaded according to the values of the Gaussian curvature. The surface becoming pointed at the comers belongs to Im z2, and the saddlelike surface corresponds to Re z2. The Gaussian curvature is -4 (1

+4IzI2)2

On the other hand, the mean curvatures of these surfaces differ:

H(Re z2) =

4 (y2 - x2) (1 + 4x2 + 4y2)3/2

,

H(Im z2) =

-8 xy (1 + 4x2 + 4y2)3/2

Now we want to interpret the formula for the Gaussian curvature expressing it as the determinant of the second fundamental form in a differentialgeometric way. To this end, consider the normal vector e3 as a map on the surface M2 with values in the unit sphere S2 C R3. The volume form dS2 of the sphere evaluated on two vectors V, W E T=S2 at the point x E S2 is the vector product

dS2(V.W) = (VxW,x). This formula is a special case of the general one for the volume form of the sphere S"'l C R" stated in §3.4, Example 28. From this, the 2-form e3(dS2) induced on M2 is easily computed: e3(dS2)(V, W) = (de3(V) x de3(W), e3)

= ((w3l(V)el +w32(V)e2) x (w31(W)el +w32(W)e2),e3)

= - w13 A w32(V, W) = -dwi2(V, W) = G dM2(V, W). We summarize the result in

Theorem 22. Let e3 : M2

S2 be the unit normal map of the surface, and let G be its Gaussian curvature. Then the induced 2 -form e3(dS2) is equal to the volume form multiplied by the Gaussian curvature:

e3(dS2) = G - dM2. Fix a point x E M2 on the surface and denote by D(x, E) the set of all points on the surface whose distance from x is less than E. Then the two-dimensional surface area of the set D*(x, e) := e3(D(x, E)) C S2 can be computed using the transformation formula for the integral as follows:

volS2 (D'(x,E)) =

J

dS2 = ±

D(x,e)

J

e3(dS2) = ±

D(x,e)

J

G C. dM2.

D(x,e)

5.4. Gaussian and Mean Curvature

163

Dividing both sides by the surface area, volM2(D(x,e)), and applying the mean value theorem of integral calculus for continuous functions, we obtain the formula volss(D*(x,e)) volA,2(D(x,e)) This was the original geometric definition of the curvature for a surface in 3IG(x)I

= £1o

space. It is-similarly to the curvature of a curve-the infinitesimal volume distortion of the normal map of the surface. For this approach to curvature, it is at first not obvious at all that the Gaussian curvature depends only on the first fundamental form of the surface. This fact forms the contents of the Theorema Egregium (Gauss, 1827). Proceeding, however, like we did here, this becomes immediate. The total Gaussian curvature of a compact, oriented surface without boundary depends only on the topology of the manifold. We prove the so-called Gauss-Bonnet formula, expressing this fact explicitly. To do so, we need a preparation. Lemma 3. Let V be a vector field of length one on the oriented surface M2, and let {el,e2} be an orthonormal frame in this orientation. Representing

V as a linear combination, V = cos r el + sin r e2, the 1 form Xv

w12 + dr

is independent of the chosen frame. The exterior derivative of Xv is

dXV = - G dM2. Proof. Every other frame el, e2 can be written as

el = e2 = and we already proved the formula w12 = w12 + dip. Moreover, r' = r - cp, and hence w12 + dr* = w12 + dr . 0 Stokes' theorem now immediately implies

Theorem 23. Let M2 be an oriented, compact manifold without boundary, and let V be a vector field of length one defined outside finitely many points x1, ... , xk E M2. If D(xi, E) C M2 are disk neighborhoods of the points xi of radius E > 0, then

r

J 12

/k

G dM2 = lim I

J

1 XvJ

5. Curves and Surfaces in Euclidean 3-Space

164

Hence we see that the integral of the Gaussian curvature is independent of the geometry of the surface. We evaluate the last formula explicitly by choosing special vector fields. To this end, consider a smooth function f : Aft -+ IR all of whose critical points are non-degenerate, and denote the number of its critical points with corresponding index by

rno (f) := the number of minima of f , the number of saddle points off,

nz z (f)

M2(f) := the number of maxima off. The vector field V = grad(f)/11grad(f) II is defined outside the set of critical

points. Computing the limits on the right-hand side of the formula, we obtain lim

0

if x; is a minimum or maximum;

2zr

JOD(z,.e)

Xv = 5 -2a if x; is a saddle point. l

To prove this, choose coordinates x, y with respect to which the function f has normal form, f (x, y) = (± x2 ± y2). We are only interested in the limit of these integrals for a 0,2 and hence we may, without loss of generality, take the metric to be flat. From gradf x a y a 11grad(f)II x2 + y2 ax 7X=2=+7y2 ay we obtain x cosy , sin r y

t

=f

x2 -+y 2

VI_X2 -+y2

Computing the differential dr leads to the formula

dr = f

x2 + y2

Here we have to take the positive sign in the cases of a minimum or a maximum, and the negative sign for a saddle point. Since

dr = ±21r, JJJSI

we obtain the limit we wanted to compute, and hence we have proved the Gauss-Bonnet theorem. Theorem 24 (Gauss-Bonnet Formula). Let M2 be a compact. oriented surface without boundary, and let f : M2 - R be a smooth function all of whose critical points are non-degenerate. Then ff2

G dM2 = 21r (rno(f) - ml(f) + rn2(f))

.

where 1110(f), MI M, m2(f) denote the number of minima. saddle points, and maxima of f. respectively.

165

5.4. Gaussian and Mean Curvature

Corollary 1. Let M2 be a compact, oriented surface without boundary. (1) The integral of the Gaussian curvature depends only on the topology of the surface, but does not depend on its fundamental forms.

(2) The alternating sum mo(f) - MI (f) + M2 (f) is independent of the choice of the function f all of whose critical points are non degenerate.

Definition 13. The Euler characteristic X(M2) of a compact surface without boundary is the number X(M12) := mo(f) - MI(f) + m2(f) Using this number, the Gauss-Bonnet formula for compact, oriented surfaces without boundary can be written as 2n

G - dM2 = X012) fExample z

n

15. The Euler characteristic of the sphere S2 is 2. In fact. the restriction of the height function h(x, y, z) = z to S2 = {(x, y, z) E 1R3 : x2 + y2 + z2 = 1) has one minimum, one maximum, and no saddle points. Thus

C d1112 = 2

1

27r

i'

a

for every surface M2 which is diffeomorphic (but not necessarily isometric) to S2.

Example 16. The Euler characteristic of the torus T2 is zero. It is not difficult to find, e.g., on the torus of revolution (see §3.1), a function with one minimum, one maximum, and two saddle points. Thus G d1112 = 0 jAf2

for every surface which is diffeomorphic to the torus.

The Gauss-Bonnet formula in its general form comprises an additional relation for every vector field V of length one with finitely many singularities. First we note that the limit lim

e-o JD(X.'0 Xv can also be computed, along the same lines as for gradient fields, for an arbitrary vector field. Start by choosing coordinates y', y2 around the point xi E A12 such that this point corresponds to y' = 0, y2 = 0. Then represent the vector field V in these coordinates,

V = V'

'

+ V2y2 .

5. Curves and Surfaces in Euclidean 3-Space

166

B

C

As before, we may assume, without loss of generality, that the metric is flat

in this neighborhood of the point, since we do not want to compute the integral itself, but only its limit. Then (Vl)2+(V2)2 = 1, and hence cosy = V 1, sin T = V2. From this we obtain - sin r dr = dV 1, cos r dr = dV2, and thus the form Xv = dr is given by Xv

= dr =

Therefore, lim

E-O

J

(x i,e)

Xv = J

(V1 dV2 - V2 dVl) := 2a Ind(V, xi). 1

This number is called the index of the vector field V at the singular point x; E M2. Notice that integrating V1 dV1 + V2 dV2 along S1 yields the same result. The geometric meaning of the index is illustrated in the figure above. The picture on the left shows some vector field (the circle is only drawn for greater clarity and has no direct relation to the vector field itself). Starting at A, move around the circle once, and assign to each point on the circle the normalized vector in S' pointing into the direction of the vector field at that point. In the case shown in this figure, one full turn in the positive direction on the left corresponds to one full turn on the right, but in the opposite direction. Hence the index of this vector field at the indicated point is -1. From the Gauss-Bonnet theorem, we obtain Theorem 25 (Hopf-Poincard). Let M2 be a compact, oriented surface urith-

out boundary, and let V be a vector field of length one defined outside {x1, ... , xk}. Then the sum k

Ind(V, X,) = X(M2) i=1

is independent of the choice of the vector field.

5.4. Gaussian and Mean Curvature

167

A further integral formula deals with the integral of the mean curvature. It goes back to Steiner (1840) and Minkowski (1900), and plays an important role in the theory of ovaloids.

Theorem 26 (Minkowski-Steiner). Let M2 C 1R3 be a compact, oriented surface without boundary, and let e3 be a unit normal field. Then

J

H(x) dM2(x) f2

= Jtz

(x,

e3(x)) G(x) dM2(x) .

Proof. Consider the three functions p2(x) = (M(x), e;(x)) on the surface M2 and compute their differentials using the structural equations: dpi = al +w12 P2+w13P3, dP2 = a2 + w21 P1 + w23 P3, dP3 = W31 P1 + W32 P2

Note also that the 1-form w := P1 W23 +P2 w31

does not depend on the choice of the tangent frame el, e2. Thus w is a globally defined form on M2. We compute the differential of w, making use of the structural equations as well as the formula for the differentials dpi. This yields

Stokes' theorem now implies the Minkowski-Steiner integral formula. The eigenvalues K1, K2 of the second fundamental form are called the principal curvatures of the surface at the point. Their arithmetic mean is the mean curvature of the surface, whereas their product coincides with the Gaussian curvature,

H=

2(i+K2),

G=

The points of a surface are divided into several types, depending on the signs of their principal curvatures. A point of the surface is called

(1) elliptic if Kl K2 > 0 ; (2) hyperbolic if K1 K2 < 0 ; (3) parabolic if Kl K2 = 0 and '4 + K2 > 0 ;

(4) flat if K1 = K2 = 0

.

Umbilic points are the points of the surface for which the principal curvatures coincide, Kl = K2. An umbilic point is either elliptic or flat.

5. Curves and Surfaces in Euclidean 3-Space

168

Example 17. The line of points with vanishing Gaussian curvature indicated in the modular surface of the function f (z) = 1/ sin z (see p. 160) consists of parabolic points. For this reason, it is called the parabolic curve. By definition, it separates the hyperbolic from the elliptic points. Theorem 27. If the surface M2 C R3 consists exclusively of umbilic points, it is part of a plane or a sphere.

Proof. By assumption, the Weingarten map is a multiple of the identity, S = x Id. The Codazzi equation,

0 = VS = V(K Id) = V(rc)

Id,

implies that K is constant. In case K = 0, the normal vector e3 is also constant. In fact, de3 = w31 el +w32 e2 = 0. Thus M2 is part of a plane. If K # 0, we obtain from r.-Id = S = de3 the equation d(e3-KM) = 0. Hence e3 - KM is a constant vector a. Thus,

Ila+KMII2 = 1, i. e., M2 is part of a sphere.

Theorem 28. A surface M2 C R3 with vanishing Gaussian curvature, G 0, and constant mean curvature, H 0 0, is part of a cylinder. Proof. The first principal curvature vanishes, r.1 = 0, and the second principal curvature is constant, K2 0 0. In an orthonormal frame el i e2 consisting of eigenvectors of the second fundamental form, we obtain the formulas W 13 = 0

and w23 = K2 a2.

Differentiate both equations: 0 = d1)13 = W12Aw23 = K2W12Aa2, 0 = w21 A w13 = dw23 = Kra da2 . Since K2 54 0, on the one hand, w12 is proportional to a2, and a2 is a closed form, dal = 0. But then dal = W12 A a2 = 0, so that altogether we obtain

dal = 0, dal = 0 and w12 = 0. The structural equations imply

de1 = 0,

de2 = r-2'0`2-e3,

de3 = K2 a2 e2 .

Hence e 1 is constant, and the surface M2 lies on a cylinder whose axis points

in the direction of el.

Theorem 29. There exists no piece of a surface M2 C R3 satisfying G =

-1 andH =0.

5.4. Gaussian and Mean Curvature

169

Proof. Let r£ 1i rc2 be the principal curvatures. Then 0

H=

(rcl + K2)

2 obtain, and -1 = G = rclrc2 immediately imply rcl = -k2 = 1. Thus we

with respect to a tangent orthonormal frame consisting of eigenvectors of the second fundamental form, the equations

w13 = al and W23 = - 02 . Differentiating these equations yields

dal =

d013 = w12 A w23 = - w12 A a2 , dal = - dw23 = - W21 A w13 = w12 A al .

On the other hand, the form w12 = A al + B a2 is a linear combination of al, 0`2 with coefficients A and B still to be determined from

dal = A al A a2 and dal = B al A a2. But then

A al A a2 = dal = -w12 A a2 = -A al A a2, B al A a2 = dal = w12 A o1 = -B a1 A a2, and hence A = B = 0. The form w12 = 0 vanishes, contradicting G =

0

-1.

Definition 14. A surface M2 c ]R3 with vanishing mean curvature, H =_ 0, is called a minimal surface. The preceding theorem states that there does not exist any minimal surface with constant, non-vanishing Gaussian curvature. Consider, more generally, the normal map e3 : lbi2 - S2 of a minimal surface. From i1 = -rc2 =

v G, we obtain the formulas W13 =

Gal, w23 = -V-v02.

The relation de3 = w31 el+w32 e2 evaluated on tangent vectors V, W E TM2 immediately implies (de3(V), de3(W))

G (V, W).

and IM2 denote the fundamental forms of the sphere S2 and the surface If M2, respectively, then the induced fundamental form (Riemannian metric) e3(IS2) is proportional to IM2:

e3(IS2) _ - G IM2

.

Thus we have proved the following theorem.

Theorem 30. The normal map e3 : M2 -* S12 of a minimal surface preserves angles, and the Riemannian metric g' := - G Ir112 defined on M2 is a metric of constant positive Gaussian curvature.

5. Curves and Surfaces in Euclidean 3-Space

170

The vector-valued position function M : M2 -R 3 of a minimal surface gives rise to harmonic functions.

Theorem 31. Let M : M2 --+ 1R3 be a surface, and let 0 be its Laplacian acting on functions. Then M2 is a minimal surface if and only if the vectorvalued function M is harmonic, 0(141) = 0. Proof. The Laplacian can be expressed by the exterior derivative and the Hodge operator (see §3.11). Using the structural equations, we compute as follows:

O(M) dM2 = d * d(M) = d * (al el + a2 e2) = d (a2 el - ai

e2)

= del Aa2+do2 el -de2Aal -dal e2 Aal - el -w12Aa2-e2

= -(w21 - el

=

Corollary 2. There does not exist a compact minimal surface M2 C R3 without boundary.

Proof. Otherwise, by Hopf's theorem (see §3.8), the components of the position function would have to be constant, a contradiction.

The normal map e3 : 1412 -+ S2 of a minimal surface with non-vanishing Gaussian curvature is a local conformal diffeomorphism. The stereographic projection from S2\{north pole} to 1R2 is a conformal diffeomorphism, too. Hence, by inverting the normal map e3 : M2 -+ S2 IIt2, every minimal surface can locally be parametrized by a map F : U -+ M2 such that (*)

OF, 49F

ayl , ayl J

=

/8F OF, 8F\ `aye aye J

and

\8F OF,OF) ayl aye

= 0.

Coordinates like these are generally called isothermic coordinates. The min-

imality of the surface expressed in these coordinates is equivalent to the requirement that F = F(yl, y2) is harmonic with respect to the Laplacian A = 82/(ayl)2 + O2/(oy2)2 on R2,

A(F) = 0. Identifying R2 with the complex numbers C, every harmonic map F : U -+ 1R3 can locally be represented as the real part of a holomorphic map 4 U -+ C3; this is a direct consequence of the Cauchy-Riemann equations and Poincare's lemma. The real partial derivatives of 4) can be expressed by the

5.4. Gaussian and Mean Curvature

171

complex derivative (z = yl + iy2), since 4' is holomorphic: a4; a4 a4 a4 ayl = az and ay2 = i az

Denoting the derivative 84'/az by W and writing 4< _ ('I'i,'1'2,'P3) with respect to the coordinates in C3, condition (*) is equivalent to the quadratic equation

'I,j+*2+q13 = 0. The solutions of this equation are described by two arbitrary holomorphic functions f (z) and g(z):

2(1 + 92(x)).

4'1(x) = f z) (1 - 92(z)), 'I2(z) = i f

13x= f(z)9(z)

Summarizing, we obtain

Theorem 32 (Weierstrass Representation of a Minimal Surface). Every minimal surface M2 C 1R3 with non-vanishing Gaussian curvature can locally

be represented in terms of two holomorphic functions f (z) and g(z). A particular parametrization is provided by the formula

F = Re(J

f 2z)

(1 - g2(z))dz, i J

L_ z) (1

+92(z))dz,

f f (z)9(z)dz).

Remark. A simply connected minimal surface always occurs as one minimal surface within an S'-parameter family; it can thus be deformed. In fact, if 4i := 4'R + i 4i1 : U C3 is holomorphic, then the equation Re(e7Q4i) = COs(a) 4iR - sin(s) 4i1 defines a family of minimal surfaces, since the quadratic equation 4 + %F2 +

i

'p3 = 0 is homogeneous.

Example 18. Very nice non-trivial minimal surfaces already arise by inserting relatively simple functions. Consider, for example, the following triple of holomorphic functions:

1-Z2 1+z2 2z2

2iz2 '

1

,

az)

satisfying the condition 'I' +'y2 +'I'3 = 0, and their complex primitives,

F = (Ft, F2, F3) = (_(z + 1/z), (z - 1/z),

In z)

2

The real and the imaginary parts of these functions determine a pair of minimal surfaces which are called adjoint to one another. Setting In z = g + ir, these are described by the formulas

FR := Re(F) _ (- sinh a sin r, sinh a cos r, r), Fj := Im (F) _ (- cosh p cos r, - cosh p sin r, q)

.

172

5. Curves and Surfaces in Euclidean 3-Space

Hence FQ = cos a FR+sin a F1 determines a 1-parameter family of minimal

surfaces joining FR to Fl. The surface FR is a helicoid, whereas F, is the surface of revolution generated by the catenary 2, the catenoid. The sequence of pictures above illustrates the deformation of both these minimal surfaces

into one another. Further examples of classical minimal surfaces can be found in the exercises.

5.5. Curves on Surfaces and Geodesic Lines Consider an oriented surface M2 C R3 in 3-space, and on it a curve 7 : M2, which we assume to be naturally parametrized. The tangent [a, b] vector t(s) to the curve is also tangent to the surface, and will be denoted by el(s). If e3(s) is the normal vector to the surface AI2 along the curve, the equation el(s) x e2(s) = e3(s) determines a third vector, which is again tangential to M2 along the curve. This way, we construct at each point of the curve an orthonormal frame, the so-called Darboux frame of the curve 2We recall that this is the curve formed by a perfectly flexible inextensible chain hanging from two supports.

5.5. Curves on Surfaces and Geodesic Lines

173

'y with respect to the surface M2. Unlike the Frenet frame, the Darboux frame of a curve in a surface is defined at every point of the curve, even at those points where its curvature vanishes. Now we decompose the curvature vector k(s) := K(s) h(s) of the curve y in R3 into the part kg(s) which is tangential and the part kn(s) which is perpendicular to the surface.

Definition 15. The vector field kg(s) is called the geodesic curvature vector, the vector field kn(s) the normal curvature vector of the curve with respect to the surface, k(s) =

kg(s) +

The vector icn(s) e3 is proportional to the normal vector of the surface with a factor Kn(s), called the normal curvature. Analogously, kg(s) := -Kg(s) e2 is proportional to e2, since the principal normal vector h(s) is orthogonal to the tangent vector. From this we obtain the geodesic curvature Kg(s) of the curve, and the square of the curvature K(s) of the curve is the sum K2(S) = K92 (S) + K2(S).

Rewriting the geodesic curvature,

Kg(S) = (kg, e2) = (k, e2) = -(del(el), e2) = -I(Veiel,e2), we are led to the following theorem.

Theorem 33. The geodesic curvature of a curve in a surface depends only on the first fundamental form of the surface. In the Darboux frame.

K9(s) = - I(Velel, e2) Similarly, the normal curvature Kn(s) of a curve can be expressed by the second fundamental form of the surface and its tangent vector:

K. = (kn, e3)

e3) =

ds

(e3

/

_ (de1(eI), e3)

Theorem 34. If y and y` are any two curves tangent at a point of the surface M2, their normal curvatures at this point coincide. Now we consider a variation of the curve y : [a, b] 1412 within the surface 1112 with fixed initial and end points. We may imagine this as a family of curves, yE (s) := y(s) + e h(s) e2(s) + O(e2),

determined by a function h vanishing at the ends of the interval, h(a) _ h(b) = 0. The length of the curve yE is

L((s)) = fbIIs)+ed(s).e2(s)+E.h(s).(s)+0(e2)II,

5. Curves and Surfaces in Euclideai 3-Space

174

and computing the derivative with respect to the parameter e yields

jb

=

(s)

e2(s) + h(s) d 2 (s)) ds .

The formulas (t(s), ez(s)) = 0 and (t(s), p(s)) = (e1, ve,ez) = w21(el) = ng(s)

then, eventually, imply

f

b

de

(L(7£(s))) Lo =

h(s) sg(s)ds.

a

Theorem 35. Let 7 : [a, b] - M2 be a curve in the surface, realizing the shortest distance between the points y(a) and 7(b) within the surface. Then the geodesic curvature of the curve vanishes,

r..9(s) = 0. Definition 16. A curve in the surface is called a geodesic line if its geodesic curvature vanishes. This condition is equivalent to VF= 0. If the geodesic curvature Kg = 0 vanishes, the curvature vector ac(s)

h(s) = k(s) = k' (s)

is proportional to the normal vector e3 to the surface. Hence a curve with non-vanishing curvature, K(s) # 0, is a geodesic line in the surface.bfz if and only if the principal normal vector h(s) coincides with the normal vector e3 to the surface,

h(s) = ±e3(7(s)) Geodesic lines realize the shortest distance between two points of the surface only if the points lie sufficiently close to one another. A longer geodesic line

does not necessarily have to be the shortest path between its end points within the surface. Examples-on S2 or on the cylinder-are easily constructed (see Example 21). We represent the curve in local coordinates y1,y2 on the surface as y(s) = (y'(s),yz(s)). If g3(y',yz) are the coefficients of the Riemannian metric and s is the natural parameter of the curve, then ;

2

1: gti(y'(s),yz(s))

dy ds

i j=1

dp' ds

=

1.

The Christoffel symbols r occur in the representation of the covariant derivatives of the basis vector fields 8/8y` as linear combinations of themselves:

z

ey y7

Erayk k=1

5.5. Curves on Surfaces and Geodesic Lines

Since

2

8

175

dyi

k

ds -

i.k=1

8 ayk ,

the following formula holds for the tangent vector t : °t{

2

-

k

ds2 +

k=1

I 1.3=1

i ds

ds

ayk

We summarize this observation in the following theorem:

Theorem 36. Let -y (s) = (y1(s), y2(s)) be a curve represented in local coordinates. It is a geodesic line if and only if the following system of differential equations is satisfied (1¢ = 1, 2): d2yk

ds2

2

+ E rjasa =o. i j=1

This is a second order system of differential equations. The general existence and uniqueness result from the theory of ordinary differential equations ensures the existence of precisely one geodesic line through any fixed point in a given direction.

Corollary 3. For each point x E M2 and every tangent vector V E T,M2 of length one on a surface without boundary, there exists exactly one geodesic line -y : (-e, e) -+ M2 satisfying the initial conditions y(0) = x, Y(0) = V.

Example 19. Let -y(s) be a geodesic line on the two-dimensional sphere S2 of radius R > 0. Since x9 = 0, the normal curvature sn and the spatial curvature a of the curve coincide. Hence

= Ian l = II(t, t) = R since the sphere consists exclusively of umbilic points. The normal vector to the sphere is e3(x) = x. Thus, on the one hand, we obtain 11

ds2)

s(k(s)) = 1

dry (s)

ds(lcn(s))

= -R ds(e3('y(s))

1

ds = - R2 r(s) and, on the other hand, the Frenet formulas for the curve imply R2

d2t(s)

d

1

t(s) + R r b(s) .

ds2 R dss) R2 Consequently, the torsion r of the curve vanishes identically. The curve -y(s)

is thus a plane curve in S2 with constant curvature 1/R, i.e., a principal circle in S2.

5. Curves and Surfaces in Euclidean 3-Space

176

Example 20. The relation K 2 = n9 + I£,2a immediately implies that every curve with vanishing curvature is a geodesic line. Formulated differently, if a straight line of R3 lies in a surface, then it is necessarily a geodesic line in this surface. Example 21. Consider the cylinder M2 in 1R3 whose axis is parallel to the unit vector a5. The normal vector at a point x E M2 is proportional to the projection of x onto the plane perpendicular to a, e3(x) =

x - (x, a)a

lix - (x, a)611

For a geodesic line -y(s) on the cylinder, we immediately obtain from h&(s) =

e3(y(s)) the equations (h(s),

= 0 and

C := (h(s), 'Y(s)) = (e3('Y(s)), 'Y(s)) = hh'Y(s) - ('Y(s), a)d1l Here C is positive and constant, since y(s) lies on the cylinder. As dt(s)/ds = K(s) hs(s) and db(s)/ds = -r(s) hs(s), the equation (h(s), a) = 0 implies

that

A := (t(s), a) and B := (b(s), a-) are constant. In case A = 0, the curve lies in an affine plane perpendicular to d and on the cylinder; hence it is a circle. For A 34 0, we differentiate the equation (hh(s), a) = 0 once, and then the Frenet formulas imply the relation

-A Ic(s) + B r(s) = 0. Differentiating the constant C2 twice, we deduce that

C ic(s) + 1 = A2. The curvature K(s) as well as the torsion r(s) are thus constant, and hence the curve y(s) is a helix (see Exercise 6). In summary, the geodesic lines on the cylinder are helices, circles, and the straight lines parallel to the 5-axis. Two points on the cylinder may be joined by infinitely many geodesic lines, only one of which is the shortest path between them:

5.5. Curves on Surfaces and Geodesic Lines

177

The equations characterizing geodesics are non-linear differential equations of second order, and hence, in general, difficult to solve. Things become a little easier if, because of the symmetry of the surface, one can find sufficiently many functions which are constant along geodesics:

Definition 17. A function f : TM -+ llt is called a first integral of the geodesic flow if f is constant on all tangent curves y'(t) of geodesics. The length of y' itself is always one non-trivial first integral, which is interpreted as energy. It expresses nothing but the fact that geodesic lines are parametrized by arc length. Further first integrals are obtained by Noether's theorem starting from isometries of the surface.

Theorem 37 (Noether's Theorem). Let V be a tangent vector field on the surface M2 whose flow 4bt : M2 -+ M2 consists only of isometries. Then the function

fv : TM -i R, fv(W) = l(W, V), is a first integral of the geodesic flow.

Proof. Consider a geodesic line y(s) on M2 in its natural parametrization. Setting W = y', the function defined in the theorem has the following derivative with respect to s: ds

(Y'(s), V(7 (s)) _ (7'(s), V(Y'(s)) + y'(s), dV(y'(s)))

The vector -y" is nothing but the curvature vector of the curve in R3, and for a geodesic line this is always perpendicular to the surface. Hence the first summand vanishes. On the other hand, the flow Ot was supposed to consist of isometries,

(W1, W2) = (d't(Wi), d0t(W2)) Differentiating this identity and evaluating the result at t = 0, we obtain

(dV(W1), W2) + (dV(W2), WI) = 0, and the second term vanishes, too, since it has to be antisymmetric in y'(s). O

Example 22. We want to find all geodesic lines on a surface of revolution. For simplicity, assume as before that the generating curve is given in natural parametrization. With respect to the coordinates s and V on the surface, a geodesic line is parametrized as y(t) _ (s(t), ap(t)) and has the first integral

E :=

I1-r'(t)II2 = s 2 + r(s)2V2

5. Curves and Surfaces in Euclidean 3-Space

178

Since the rotation through the angle V around the z-axis is an isometry of the surface, the vector field 8/8W satisfies the assumptions of the Noether theorem and gives rise to a further first integral,

M := (y', 8/8V) = r2c . The existence of this second invariant is known as Clairaut's theorem (1731) and has the following geometric interpretation. Consider a geodesic line

through a point of the surface, and also the meridian 77(t) = (so, W(t)) through this same point. The angle a formed by the geodesic and the meridian is computed as

cos(a)

_

('y', W) 11Y11 ' 11741

r2Vt2

E r2Vp'

rI

E

Therefore, M is constant if and only if r(s) cos(a) is constant. For the qualitative discussion of the geodesic lines, we take E = 1. Inserting yY _ M/r2 into E = 1, we obtain

s 2r2 = r2 - M2, thus 0 < M2 < r2 . If M vanishes identically, then gyp' = 0, and p has to be constant. This corresponds precisely to the generating curve of the surface of revolution; each profile curve of F is thus a geodesic. In the other extremal case, M2 = r2, we have s' = 0, i. e., a has to be constant. This curve is a meridian on F; note, however, that in this case differentiating M2 = r2 immediately leads to r r' = 0, which, in turn, implies r' = 0. A meridian of F is a geodesic if and only if the radius function r has an extremal point there. All that remains is the generic case, 0 < M2 < r2. The condition E = 1 is equivalent to

a' = flr r2-M2. In order to eliminate the curve parameter t, we view s as a function of gyp; hence ds/dcp = s'/V, d,p

= f M r2 - M2,

and thus rSOP)

1

MJ

,o

r(s)

r(s) -

r,v

ds=J do =fop. M2 0

This provides a parametrization of the geodesic lines in the case of surfaces for which the integral on the left-hand side is elementarily integrable and can be solved for s.

5.5. Curves on Surfaces and Geodesic Lines

179

Example 23. Let 7.(2 be the hyperbolic plane (see Example 9). We again parametrize the geodesic ry as 'y(s) = (x(s), y(s)). Then the condition that -1 is parametrized by arc length takes the form xi2

y2

i2

+ y2 = 1 .

Since the metric does not depend on x, translations in the x-direction are isometries; the associated vector field is 8/8x. Hence

M :_ (y', 3/ex) =

lye

is a first integral. If M vanishes, x has to be constant, and the geodesic lines are precisely the open straight half-lines parallel to the y-axis. In the general case, rewrite the first condition, inserting M. A short calculation leads to y = ±y l - M2y2. Again, we view y as a function of x and set r:= 1/jMI, dy = y dx x'

1

r2 _ y2

y

Directly integrating this equation and denoting the integration constant by a yields

-

r2 - y2 =

(x - a)

hence (x - a)2 + y2 = r2.

The trace of the geodesic line is thus a semi-circle whose center lies on the x-axis.

Once again we now turn to the Gauss-Bonnet formula. Let M2 be an oriented, compact surface with boundary 9M2, and let V be a vector field of length one defined on M2 except at finitely many points xi, ... , xk in the interior of M2. Assume, moreover, that V is perpendicular to 9M2 at the boundary 3M2 and outwardly directed. Consider the 1-form Xv introduced in §5.4. From the formulas proved there, we deduce aM2

J

dXv =

J

k

Xv - 27r

Ind (V, xi).

Near the boundary 8M2, we choose an orthonormal frame el, e2 such that el = V. Then Xv = w12, and e2 is the unit tangent vector to the boundary. Since W12(e2) = I(Dezel, e2) = -I(el, V2e2) = K9, we obtain the GaussBonnet formula for manifolds with boundary. Theorem 38 (Gauss-Bonnet Formula). Let M2 be a compact, oriented surface with boundary, and let V be a vector field of length one defined on M2

except at finitely many points xl, ... , xk in the interior of M2. Assume,

5. Curves and Surfaces in Euclidean 3-Space

180

moreover, that at the boundary V is perpendicular to the boundary and outwardly directed. Then

f

J

G dM2 +

,112

f icy d(8M2) = 2a J0M2

k

Ind(V, xi). i=1

This has similar consequences as in the case of surfaces without boundary. On the one hand, the sum of the integrals f1112 8

IMZ

K9.d(aM2)

is independent of the first fundamental form. On the other hand, the sum of the indices of a vector field does not depend on the vector field.

5.6. Maps between Surfaces The length IIVII =

I(V, V) of a tangent vector can be determined by

measuring the length of curves. To see this, choose a curve -y on the surface J112 in arbitrary parametrization whose tangent at t = 0 coincides with V, y(O) = V. The length of the curve segment y([0, t]) is defined as

L(t) = f0

I("Y(Ft),'Y(p))dp ,

and computing the derivative at t = 0 yields dL(t) I(V, V) . dt t=o The first fundamental form of the surface is thus completely determined by M2 the lengths of curve segments, and vice versa. Hence each map f : M2 between two surfaces which does not alter the lengths of curve segments has a differential df : TM2 -+ TM2 preserving the first fundamental forms I,1f2 and IAt2,

I (IT12) = IJ12.

Definition 18. An isometry f : M2

M2 between two surfaces is a

length-preserving smooth map.

The Gaussian curvature of a surface is a geometric quantity depending on the first fundamental form. This immediately implies

Theorem 39. Let f : M2 -+ M2 be an isometry. Then the Gaussian curvatures G,2, Gh12 at corresponding points coincide:

GM2(x) = Gh12(f(x)) Corollary 4. There does not exist an isometric map from an open subset of S2 onto an open subset of the euclidean space R2.

5.6. Maps between Surfaces

181

For this reason, cartography was forced to look for other suitable maps. Especially important are conformal (angle-preserving) and volume-preserving AI2 is called maps. Let us first turn to the former ones. A map f : M2 conformal (or angle-preserving) if the angle between two intersecting curves 11,12 in M2 is the same as that of the image curves f o 11, f o y2 in R12. These maps can be characterized using the first fundamental form.

Theorem 40. A map f : M2 11 I2 is conformal if and only if there exists a positive function h : M2 - (0, oc) such that f `(ITf2) = h I tif2. 2 Proof. This is an easily proved fact from linear algebra. Let ( , ) 1 and be two positive definite scalar products on a two-dimensional vector space satisfying the equation

(V, W)1

(V, W)2

IIVII11IWII1

IIVII2IIWI12

for all vectors V. W. Then there exists a positive number h such that

(V, W)2 = h (V, W)1 . We already encountered conformal maps. The normal map e3 : M2 _ S2 of a minimal surface in R3 is conformal (Theorem 30). Complex analytic functions are conformal maps from the plane R2 to itself. In fact, if f = (u, v) = u + i v is a map f : 1R2 1R2, then the following formula holds for the first fundamental form IR2 = dx2 + dy2:

f'(Ia2) = (utdx+ul,dy)2+(v.dx+vydy)2 = (u2 + v2)dx2 + 2(uruy + v=vy)dx O dy + (v2 + vy)dy2.

The Cauchy-Riemann equations, ur = vy, uy = -vi, imply that, for the induced form,

f(1R2) =

If,(z)12.Ia2.

Hence every complex-differentiable function f = u + iv is a conformal map, and the square of the absolute value of its derivative, I f'(x)12, appears as the conformal factor in the sense of two-dimensional geometry. The best-known conformal map is the stereographic projection from the two-dimensional

sphere to the plane. This shows that conformal maps from R2 to S2 exist, and so angle-preserving geographic maps can be drawn.

Example 24. In coordinates, the stereographic projection f : R2 the map 2x

2y (l+x2+y2' 1+x2+y2'

1 - x2 - y2 1+x2+y2).

S2 is

5. Curves and Surfaces in Euclidean 3-Space

182

The partial derivative vectors of f are Of

2

Ox - (1+x2+y2)2 Of

(1 - x2 + y2, -2xy, -2x)

,

2

-

(I + x2 + y2)2 (-2xy, 1 + x2 - y2, -2y)

and hence

Of Of

\ 8x' 8x

_

4

_

Of Of

8y' ay

(1 + x2 + y2)2

and

8f Of

8x' 8y

_

0.

These formulas show that the stereographic projection is conformal.

Example 25 (Mercator projection). Let (gyp, tai) be spherical coordinates on the sphere S2\{N, S} outside the north and the south pole (see §3.2, Example 14). Define a map from the sphere S2 to R2 by the formula

log (tan(4 + 2111 This turns out to be a conformal map with an additional interesting property: curves of constant azimuth (i.e. forming constant angles with the meridians on S2) are mapped to straight lines under f. For this reason, the Mercator projection (1569) has been one of the most widely used methods for the creation of nautical maps for centuries. The reader may find an extensive discussion of further important map projections in the book by (V,

K. Strubecker (Volume 2) cited in the bibliography. Another interesting class of maps between surfaces are those preserving the two-dimensional volume (surface area) of measurable subsets of the surface. Such a map f : M2 --+ M2 is characterized by the property that the volume form is preserved, f*(dM2) = dM2.

We will call such an f an area-preserving map. Isometries are the maps which are both conformal and area preserving.

Theorem 41. A map f : M2

M2 is an isometry if and only if f is

conformal and area-preserving.

Proof. The relation f*(IM2) = h. IM2 implies f*(dM2) = h2d.112' and hence also the assertion.

The sphere S2 (or pieces of it) thus cannot be mapped to the plane in an angle- and area-preserving way. There exist, however, angle-preserving maps, as the stereographic projection shows. We now want to describe an example for an area-preserving map from S2 to 1R2.

5.7. Higher-Dimensional Riemannian Manifolds

183

Example 26. Delete the north and the south pole from the sphere S2, join them by a straight line, and consider the cylinder whose axis this line is:

Z = {(x,y,z) E R3: x2+y2 = 1}. To each point different from the north and the south pole, P 0 N, S, there corresponds one straight line containing the point P which is perpendicular to the fixed axis. This line intersects the cylinder in two points, and we denote the one closer to P by f (P). This defines the so-called Lambert projection,

f : S2\{N,S} -> Z. Composing this with an isometry from the cylinder to the plane, we obtain an area-preserving map from S2 to R2.

To see this, let el, e2 and ei, e2 be the orthonormal frames on S2 and Z. respectively, indicated in the above picture. Then an elementary geometric argument immediately leads to the relation

df(ei) = rei, df(e2) = Tee Here r is the radius of the meridian of S2 tangent to e2. These formulas prove that f is an area-preserving map.

5.7. Higher-Dimensional Riemannian Manifolds Following §3.11, we will conclude this chapter by a brief survey of higherdimensional Riemannian manifolds. Only the basic facts are discussed, and no comprehensive introduction into Riemannian geometry is intended. As in the case of surfaces, every pseudo-Riemannian manifold (11I'", g) can be endowed with a covariant derivative V of vector fields with respect to vector

5. Curves and Surfaces in Euclidean 3-Space

184

fields which is uniquely determined by two conditions. This formalizes the notion of parallel displacement of vectors on a manifold. In fact, LeviCivita's idea (1916) that every Riemannian space admits a unique parallel displacement preserving the metric turned out to be fundamental for the further development of geometry. Theorem 42. Every pseudo-Riemannian manifold has precisely one covariant derivative V W of vector fields with the following properties:

(1) V (W1 + W2) = V (Wi) + V (W2) (2) Vv, +v2 W = Vv, W + Vv, W ;

(3) if f is a smooth function, then

Vv(f-W) =

and V f.vW = f VvW ;

(4) Vg(WI, W2) = g(VvW1, W2) +g(W1, VvW2) (5) VvW - VwV = [V, W). The vector field VvW is determined by the expression

2g(VuV, W) = U(g(V,W))+V(g(U,W))-W(g(U,V))+g([U,V}, W) +g(V, [W, 141) - g(U, [V, W])

Proof. We prove uniqueness of the covariant derivative and derive in particular the stated formula for VvW. By assumption, we have

U(g(v, W)) = g(VuV, W) + g(V, V W) , V(g(U, W)) = g(VvU, W) + g(U, V W), W(g(U, V)) = g(VwU, V) + g(U, V WV).

Because of the properties of V, we may introduce the commutators of the vector fields into these equations:

W(g(U,V)) = g(VuW,V)+g(U,VvW)+g([W,U],V)-g(U, [V, W]), V(g(U,W)) = g(VuV,W)+g(U, VvW)-g(W, [U,VI) From the first, fourth, and fifth equation we obtain the formula for the scalar

product g(VuV, W). Hence there exists at most one covariant derivative with the required properties. Conversely, the formula determines the vector VvW, and an analogous computation shows that the operation thus defined is a covariant derivative with all the properties needed.

Definition 19. The covariant derivative V W is called the Levi-Civita connection of the pseudo-Riemannian manifold.

5.7. Higher-Dimensional Riemannian Manifolds

185

Let V1, .. . , Vm be an arbitrary frame of vector fields defined on an open subset of Mm. Representing the covariant derivative of the basis vector

fields as EU]

OV,Vj

17

13

- Vk,

k=1

we introduce the Christoffel symbols of the Levi-Civita connection. The symmetry properties of the functions r strongly depend on the chosen frame. If the vector fields Vi mutually commute, [Vi, Vj] = 0, and if gij = g(Vi, Vj) are the coefficients of the Riemannian metric, then formula (5) from Theorem 42 simplifies, and we obtain the expression m

rj =

9>9'(Vi(9j0)+Vj(9ta)-Va(9ij)).

a=1

In this case, the Christoffel symbols are symmetric in the lower indices,

r =r

(for commuting frames).

If the frame V1 := el, ..., Vm := em consists of mutually orthogonal vector fields of constant length ±1, then the equality

0 = Vi(g(Vj,Vk)) = 9(VV,Vj,Vk)+9(Vj,VV.Vk) implies the following symmetry property of the Christoffel symbols:

r = -rk

(for orthonormal frames).

Example 27. The first fundamental form of a surface of revolution can be written (see Example 3) as I = ds2 + r(s)2 dcp2 . With respect to the commuting frame V1 = 8/8s, V2 = 8/8cp we obtain, for the "classical" Christoffel symbol rte,

I'l 22

2

r,g =8.;

aQ

I

p9 + a-g ,0 -

-r

8 as9"''J

[

2

as )2 J

r'

and similarly rig = r21 = r'/r. The other Christoffel symbols vanish. From Example 3 we take the formula W12 = (r'/r) 02 with respect to the orthonormal frame el = 8/8s, e2 = (1/r) 8/8cp and its dual frame ?1,02 considered there. Together with the definition of the "Cartan" Christoffel symbols,

W12 = r11 ai + r21 o2 = 0.01 + -a2,

this yields r2 1l = 0 and r2, = r'/r. Similarly we have W21 =

r11 2 = 12 o1+r1 22 a2

r

5. Curves and Surfaces in Euclidean 3-Space

186

implying r12 = 0, r22 = -r21. Again denoting by Q1, the 1-forms dual to the frame e1.... , c,,,, we introduce the local connection forms

wij := Ej -9(Vei,ej) = oj(Vei) with Ej :=g(ej,ej) By definition, they satisfy m

wij =

rQi - va

and

Ei - wij + Ej wji = 0 .

a=1

N%,e compute the exterior derivative doi using the properties of the LeviCivita connection:

dui(V,W) = V(Qi(W)) - W(oi(V))

-oi([V,WI)

M

_ Eej-{9(Vvei,ea).a.(W)-9(Vwei,ea)-a.(V)} a=1

M

_ 1: aQAw,,i(V,W) a=I

This leads to the first structural equation of a pseudo-Riemannian space. In order to formulate the second structural equation, denote by SZ := (wi j ) the antisymmetric (m x m) matrix composed of the 1-forms 4;ij. This additional structural equation computes the matrix d!2 - S1 A S2 of 2-forms, and

to formulate it we need the curvature of the space. The curvature tensor 7 (U. V)W := VuVVW-VVOuW-V1U,VIW of a pseudo-Riemannian manifold is introduced generalizing the case of surfaces, and all the identities of Theorem 15 in §5.3 remain valid. In particular, the first Bianchi identity holds:

R(u, V)W + R(V,W)u + R(W,u)V = 0. However, in the higher-dimensional case the curvature tensor does not reduce to a single function. By definition, it is a (3,1)-tensor. This can be transformed into a (4, 0)-tensor, the Riemannian curvature tensor: R(U,V, W, Wi) := 9(R(U, V)W, W1) . Referring to an orthonormal frame e1, ... , e,,, of basis vector fields of lengths ±1, we obtain from m

Rijk el

R(ei, ej )ek 1=1

the components Rijk of the curvature tensor. These can be computed by the formula Rtijk = El . R(ei, ej, ek, el) from the Riemannian curvature tensor.

5.7. Higher-Dimensional Riemannian Manifolds

187

Theorem 43 (Structural Equations for Pseudo-Riemannian Manifolds). Let el, ... , em be a local orthonormal frame on a pseudo-Riernannian manifold, and let o1, ... , an be the dual frame. Then the following equations hold: Lul

(1) dai = 1: a. A wai ; Q=1

m

(2) dwij =

1

win A waj + a=1

2

m as A a3 . a,J3=1

Proof. We already derived the first equation. To prove the second, we use the properties of the Levi-Civita connection and obtain, after several elementary steps,

dwij(V,W) = a (R(V, W)ei) +Ej - (g(Vwe;, Vvej) - g(Vvei, Vwej)) 71'

ejEa(Qa(Vwei)Qa(Ovej)

_ aj(R(V, W)ei) + a=1

- aa(Vwej)an(Vvei))

Moreover,

-wnj(V)

ej - e -an(Vvej) = and, similarly, ej en aa(Vwej) _ the proof,

(W). Altogether, this completes

m

winAwaj(V.W)+oj(R.(V.W)ei)

dwij(V,W) _ a=1

_

wiaA waj(V,W)+RVWi

0

a=1

Corollary 5. If the curvature tensor of a pseudo-Riemannian manifold X11"' vanishes, then in a neighborhood of each point there exists a chart in which the coefficients of the Riemannian metric are constant,

g = diag(±1, ....±1). Proof. First choose a local orthonormal frame e1, ... e.. and consider the matrix n = (wij) of connection forms. Then dfl = SZ A Q, since the curvature tensor vanishes. By Theorem 6 in §4.3. there exists a

(pseudo-)orthogonal matrix A of functions such that

ft = dA A-1 and A(O) = diag(± 1.... , ±1).

5. Curves and Surfaces in Euclidean 3-Space

188

Denote by r := (a1 ... a,,,) A a special system of 1-forms determined by a1, . . . , o,,, and the matrix A. The structural equations imply that this form is closed.

dr =

0.

Hence, by Poincare's lemma, each 1-form r, can locally be represented as the differential of a function fi. This way, we construct local coordinates on the manifold, and for the pseudo-Riemannian metric g we obtain the expression

g = ±ai + ... t a2, = ±rl + ... f T'2 = ±(dfl )Z + ... f (dfm)Z. Definition 20. A pseudo-Riemannian manifold with vanishing curvature' tensor is called flat. The preceding corollary thus states that every flat pseudo-Rieinannian space is locally isometric to the standard space Rm.k with the following metric of index k: dxm-k dXin-k dxm-k+l dXm-k+l - ... - dx"' ) dx'n. dxl &I +... +

,

-

Consider a curve -y(t) in the pseudo-Riemannian manifold (111',g) and a vector field W(t) along this curve. In addition, we may suppose that locally the latter is the restriction of a vector field W defined on Mm. In the neighborhood of a point on the curve we choose the coordinates yl.... , y'n and. represent the vector field W and the tangent vector i(t) in these coordinates:. m

m W = ` yi.

yi ^'y(t) = E^"`(t)

i=1

i=1 m

d

A

E1'(t) ayi ('Y(t)) i=1

Then

(07(r)W)(y(t)) = > k=1

dWA(7(t)) +

dt

['

ii(t) . W'(7(t)) . q(7m)

8 8yk

Along the curve y(t), this vector field is independent of the particular (lo-cal)

extension of the vector field W(t). originally defined only on y(t), to a vector field W. The formula thus uniquely defines a vector field VW/dt,l the covariant derivative of W(t).

Definition 21. A vector field W(t) is called parallel along y(t) if

vW

dt

0.

5.7. Higher-Dimensional Riemannian Manifolds

189

In coordinates, this condition is equivalent to the following system of ordinary differential equations: dWk

o.

14#

The existence and uniqueness theorem for ordinary differential equations ensures the existence of a unique parallel vector field W (t) for every initial vector W. E TAi a) at the initial point y(a) on the curve,

vw dt

= 0. W(a) = Wa.

The value LV(b) at the end point ';f(b) of the curve is called the parallel displacement P., of the vector W. along the curve y(t),

P,(lV) := IV(b). Because of the linearity of the differential equation and the property

U(g(V,W)) = g(V V,W)+g(V,vuW) of the Levi-Civita connection, the parallel displacement P.7:T.(,.)Alm

Try(b)A.lm

is a linear isometric map between tangent spaces. The parallel displacement P.y,..Y2 along a curve formed by combining two curves y1 and y2, for which the end point of the first coincides with the initial point of the second, into a single curve y1 * 72 leads to the same result as the superposition of the single parallel displacements, P12 o P,., = P'Yl *'12

Consider a surface (1112, g) with a positive definite Riemannian metric and a closed curve y in M2 bounding a domain N2 C 1112. Parallel displacement along y then is a rotation in the tangent space T.,(o)M2. The corresponding rotation angle can be computed from the Gaussian curvature.

Theorem 44. If the closed curve y bounds a domain N2 in a surface AI2 with Gaussian curvature G, the parallel displacement along the curve is the rotation through the angle

0=

JN2

r

Proof. Choose an orthonormal frame e1,e2 along -y in which e1 is tangent to the curve. In any parametrization y : [0, L] Ail of the curve, a vector field V = V1(s) e1(s) + V2(s) e2(s) is parallel if dV1 ds

= w12(e1) . V2

and

dV2 ds

= -w12(el) V1.

5. Curves and Surfaces in Euclidean 3-Space

190

Consider the complex-valued function z : [0, L] C defined by z(s) := Vl(s) + i. V2(s). Then we can write this system of ordinary differential equations in the form dz

Z

/

= -2 W12(el) z.

The solution of this is

z(8) = z(0) exp l _ i

e

Wl2(el) dsJ

J0 and, in particular, for the rotation angle of the parallel displacement we obtain

0 = - fr w12 =

JN2 d'012 =

JG.dM2.

Example 28. It is a peculiar property of the hyperbolic space 7.12 that the hyperbolic angle coincides with the usual euclidean angle. In general, the angle does not change under parallel displacement. Hence the parallel displacement of a vector along a geodesic line in a surface is the vector at the end point of the line forming the same angle with the tangent vector as the original vector did at the initial point. The points PQR in the picture below are the vertices of a geodesic triangle in the hyperbolic plane (see

Example 23). Start at P with any tangent vector (1), translate it to the point Q (2), then along the upper geodesic arc to R (3), and finally along the straight segment RP back to P (4). The resulting vector obviously does not coincide with the original one.

Parallel displacement allows one to generalize the covariant derivative V VT with respect to vector fields to arbitrary tensors T. We explain this for tensors of type (k, 0). Assume that at a point a E Mm, the vectors W1.... W. E

5.7. Higher-Dimensional Riemannian Manifolds

191

TXM' are given, and let y(t) be the integral curve of the vector field V passing at time t = 0 through the point x E Mm. Then we define

d

(VyT)(WI, ...,)4)(x) :=

at(Ty(t)(Py(WI),

...,P7(Wk))It=o'

Here Py denotes parallel displacement along the curve y from the point x = y(0) to the curve point -f(t). If, for example, T is a (1, 0)-tensor and W a vector field, we obtain the formula

(VyT)(W)(x) = V(T(W))(x) - T(VyW)(x). Theorem 45. The covariant derivative VyT of a (k, 0)-tensor is given by

(VvT)(Wi, ..., Wk) = V(T(Wi, ... , Wk)) - T(VvW1, W2, ... , Wk) - ... - T(W1i ... , Wk-1, VVWk) . Remark. The metric g, viewed as a (2, 0)-tensor, is parallel:

(Vy9)(Wl, W2) = V(9(Wi, W2)) - 9(VvW1, W2) - g(WI, VvW22) = 0.

A completely elementary but somewhat lengthy computation, using the properties of the Levi-Civita connection and the definition of the curvature tensor, leads to an identity for the covariant derivative VTZ of the curvature tensor, the so-called second Bianchi identity. We omit the simple proof and formulate only the result. Theorem 46 (Second Bianchi Identity). Let U, V, W, W1, W2 be five vector fields on the pseudo-Riemannian manifold (M', g). Then (VuR)(V, W, W1, W2)+(Vy7)(W,U, W1, W2)+(VWR)(U, V, W1, W2) = 0.

Define the length of a curve y : [a, b] -+ Mm in a pseudo-Riemannian space by the integral

L(y) :=1 b a

I9( (t), ry(t)) I dt .

For a variation yE : [a, b] - Mm of the original curve y(t) with fixed initial and end points, y. (a) = y(a),-yy(b) = y(b), the field of the variation, W (t)

(yE (t)) I E=o,

is a vector field defined along the curve y(t) vanishing at the end points. Its covariant derivative VW/dt along the curve y(t) coincides with the covariant derivative Vyy(t)/de of the vector field ye(t) defined along the curve i7(c) -Y" (t),

dt (t)

-

t)

V de

L-0

5. Curves and Surfaces in Euclidean 3-Space

192

This formula is an immediate consequence of the local expressions for the two covariant derivatives. Parametrizing the curve y(t) by are length. we obtain the following formula for the variation of the length of the curve:

d(L(,E))IE_o =

J

g(y

,y(t))dt

t) 1==0

=

=

jbg(dw(t),ry(t))dt

lb g (W(t), 7- (t))dt.

Theorem 47. If the curve y : [a, b] Mm is a critical point of the length functional defined on the set of all curves with fixed initial and end points. then the tangent vector y: (t) is parallel along the curve y(t). V7 0. dt =

Definition 22. A curve y(t) in the pseudo-Riemannian space (M"'.g) whose tangent, vector field is parallel is called a geodesic line.

Remark. If the curve y(t) = (y' (t), ..., ym(t)) is given in local coordinates, the equation for it to be a geodesic line is d2yk dt2 +

M dy' dy'

E

*a=1

dt

-;it

r

= 0.

This is an ordinary differential equation of second order, and Corollary 3 from §5.5 holds correspondingly:

Corollary 6. To each point x E M' and every vector V E TIMtm on a pseudo- Riemannian manifold without boundary there exists precisely one geodesic line y,; : (-e, e) -+ Mm such that -y, (0) = x and i,(0) = V.

The geodesic line y (t) thereby determined is not necessarily defined for all values of the parameter t E R. The subset Ex C TxA1t of all vectors V E TxMt for which exists for t = 1 is an open neighborhood of the zero vector in TTM". The exponential map at the point x of the pseudoRiemannian manifold maps Ez to the base space, Expx :

Ex -p Mm,

Expx(V) = yv(1)

The exponential map is smooth, and its differential d(Expx) : To(T1M'") _ T,Al"' -. TZMm is the identity,

dExpx(W) = W. dt(Exp(tW))It=o = dt(y'.w(1))It=" = Hence the exponential map Expx is a diffeomorphism from an open neighborhood 0 E Ux C TZMm of the zero vector in the tangent space onto an open neighborhood x E Vx C M^' of the point in the base space.

5.7. Higher-Dimensional Riemannian Manifolds

193

Simpler curvature invariants of the pseudo-Riemannian manifold arise from the curvature tensor by contraction (taking the trace), the most important ones being the Ricci tensor and the scalar curvature. The sectional curvature

appears as the Gaussian curvature of subplanes of the tangent space. l e shall now discuss these curvature quantities.

Definition 23. The Ricci tensor Ric is the first contraction of the curvature tensor, m

Ric(U, V) :=

F'g'"R(ei, U, V, e3) .

ij=1

The algebraic identities of the curvature tensor imply that Ric is a symmetric (2,0)-tensor, Ric(U, V) = Ric(V,U).

Taking the trace of the Ricci tensor with respect to the pseudo-Riemannian metric g (§5.4) leads to the scalar curvature r of the space.

Definition 24. The scalar curvature of a Riemannian manifold is the function

T ._

ij9k1R(ei, ek, el, e3) i j.k.l=1

Contrary to the surface case (m = 2), the curvature tensor is no longer determined by the scalar curvature for manifolds of dimension at least three. Thus the Ricci tensor and the scalar curvature do not contain all the curvature information about a manifold. The last of the curvature quantities is the sectional curvature. Apart from the point of the manifold it depends, in addition, on a non-degenerate 2-plane in the tangent space. This is a two-dimensional subspace E2 C ,,Mm whose orthogonal complement.

(E2)1 = {W E T Al- : g(V,W) = 0 for all V E E2}, is complementary to E2, E2 e (E2)1 = TXM. Intersecting E2 with an open set 0 E U= C T=Ai"` on which the exponential map is defined and a diffeomorphism, we obtain a surface in M"',

M2(E2) := Expr(E2 n U:) .

Its tangent plane at the point x is E2, and, by restriction, the pseudoRiemannian metric g of M'" induces a metric on M2(E2). The Gaussian curvature G(x) of the latter at the point x E h12(E2) is called the sectional curvature K(x. E2) of the pseudo-Riemannian manifold at the point x in the direction of the 2-plane E2. It can be computed using the curvature tensor of Af '.

5. Curves and Surfaces in Euclidean 3-Space

194

Theorem 48. Let V1, V2 be a basis of the non-degenerate plane E2. Then

K(x, E2)

--

1Z(V1, V2, V2, Vl)

g(Vl, V1) g(V2, V2) - g2 (VI, V2)

Proof. For the proof, we assume that the metric of MI is positive definite. This is no essential restriction, but some of the formulas simplify. Note first that the right-band side of the formula for K(x, E2) does not depend on the choice of the basis V1, V2; thus it suffices to prove this formula for an orthonormal basis in E2. Fix an orthonormal basis e1(x), e2(x), ... , e,,,(x) at the point x E C Al" such that the first two vectors belong to E2. We extend this basis to a local orthonormal frame by parallel displacement along the geodesic straight segments {Expx(t W) : W E TXMm}. At the point x this frame is parallel,

Vei(x) = 0. Furthermore, on 1612(E2) the vector fields el, e2 are tangent to this surface, since M2(E2) consists of geodesic straight segments with initial vectors in E2. Let an, w,j be the forms dual to the frame e1, ..., e,,, and the connection forms of the Riemannian manifold Mm, respectively. The embedding f : M2(E2) Mm allows us to restrict these forms to the surface,

A. := J'(aa), ti.i

f'(wij)

Then µ3 = p4 = ... = um = 0, and the structural equations of M'" imply, after restriction to M2(E2), that d{ll = ll2

t21,

dA2 =

n t12,

t In A Cat + n121 pI A 142Sln

X12 = a

The pair µ1,/l2 is, within the surface M2(E2), the frame dual to el, e2, and the last equations show that the connection form w12 of M2(E2) Coincides with t;12, wit = e12. The Gaussian curvature a of M2(E2) is determined by

- G - JAI A P2 = &w12 = d512 = At the point x E M2(E2), however, we have hence we obtain

- G(x) =

R2121

tlnAta2+R2121 p1 AP2 = g(De,e1, ea) = 0, and

= g(R(el, e2) el, e2) = - g(7Z(el, e2) e2, e1).

Special Riemannian manifolds are the so-called Einstein spaces and spaces of constant sectional curvature.

Definition 25. A pseudo-Riemannian manifold is called an Einstein space if the Ricci tensor is proportional to the metric, Ric = h g.

5.7. Higher-Dimensional Riemannian Manifolds

195

Remark. Contracting the equation Ric = h g leads to r = m h, i. e., the factor of proportionality is uniquely determined by the scalar curvature of the manifold, T

Ric = m _19. We prove that this factor is constant. Theorem 49. If (Mm. g) is a connected Einstein space of dimension m > 3. then the scalar curvature T is constant.

Proof. Choose an orthonormal frame el, ...,en and an arbitrary vector field U. After we insert the vector fields V = W2 := ei, W := ej, and W1 ej into the second Bianchi identity and sum over all indices 1 < i, j < m, an elementary computation leads to the equation

> ej ((Vu)Ric(ej, ej) - 2(V Ric)(U, ei)) = 0.

i

If the Ricci tensor is proportional to the metric, Ric = h g, the relation Vg = 0 implies VuRic = U(h) g, and the last equation becomes

m U(h) - 2E ejej (h)g(U,ei) = 0. Considering the vector fields U := ek, this yields (m - 2) dh(ek) = 0, and, in case m > 2, the differential dh of the function h vanishes. 0

Example 29. In spherical coordinates, the standard metric of the twodimensional sphere is the quadratic form gJs2 = COS2(71') djl2 + d/12

.

Choose a pseudo-Riemannian metric h in the product R+ x R and a function f : R+ x R R. On the four-dimensional manifold (1R3 -0) x lR = S2 x R+ x R we can then consider the rotationally symmetric metric

g := h (B e2f(r,t) .

gJs2 .

A longer computation reveals that g is a four-dimensional Einstein metric with vanishing scalar curvature if and only if the following pair of differential equations is satisfied:

e-2f = A(f)+2lgrad(f)I2 and

Odf.

Here, A and grad are the Laplacian and the gradient of the two-dimensional pseudo- Riemannian manifold (R+ x Ift, h), G is its Gaussian curvature, and

Hess h(f)(V, W) := h(Vy(grad(f )), W)

5. Curves and Surfaces in Euclidean 3-Space

196

is the symmetric Hessian form of the function f with respect to the metric h. Choosing, for example, the metric

h=

Ll +

2A1

r

dr2 + c J

r

r

dr C dt - c2 I 1 111

2Af 11 dt2 r J

and the function f (r, t) = log(r) gives rise to the Schwarzschild-Eddington metric on (R3 - 0) x R, which, in Einstein's general theory of relativity, models the gravitational field generated by a mass concentrated at the point x = 0. Further solutions of this system of differential equations describe the gravitational field of a rotating mass (Kerr metric).

Definition 26. A pseudo-Riemannian manifold (Mm,g) is called a space of constant sectional curvature (or space form) if there exists a function K' : 111"' -+ R such that for every 2-plane E2 C TXMm

K(x, E2) = K'(x). Theorem 50. The sectional curvature K = K' of every space Mm of dimension m > 3 is constant, and the curvature tensor is algebraically determined by the metric,

R.(U,V)W = K'(g(V,W) -U -g(U,W) -V). Proof. Consider the tensor S(U, V, W1, W2)

= R(U, V, W1, W2) - K* (9(V, Wi)9(U, W2) - 9(U, W1)9(V, W2)),

and note that S has the same symmetry properties as the curvature tensor:

(1) S(U,V,W1,W2) = -S(V,U,W1,W2). (2) S(U, V, W1, W2) = -S(V,U, W2, W1) .

(3) S(U,V,W1,W2)+S(V,W1,U,W2)+S(W1,U,V,W2) = 0. (4) S(U, V, W1i W2) = S(W1, W2,U, V) .

An additional property of the tensor S results from the formula for the sectional curvature:

(5) S(U, V, V, W) = 0 .

A calculation only making use of equations (1) - (5) shows that this tensor has to vanish, S - 0. In fact, from 0 = S(U. V + W2, V + W2iU) = S(U. V. V, U) + S(U, V, W2, U) + S(U, W2, V, U) + S(U. W2, W2, U)

= 2 - S(U, V, W2, U)

5.7. Higher-Dimensional Riemannian Manifolds

197

we first conclude that S(U, V, W2, U) = 0. Hence we have succeeded in replacing the third argument. V in the identity (5) by an arbitrary vector W2. We repeat this step once more, starting from S(U, V, W2, U) = 0. Thus, as the fourth entry another independent vector W1 can be substituted for the vector U, that is.. S vanishes, and the curvature tensor of a space form is determined by

R(U, V)W = K`(g(V, W) U - g(U, W) V). Contracting the curvature tensor, the resulting Ricci tensor is proportional to the metric,

Ric = (m-1) K*.g. Hence M"' is an Einstein space, and K* is a constant function.

0

The components Ra31 of the curvature tensor of a space form with positive definite metric (for simplicity) are

Ra3i = K' - (b3idaj - 63j6ai), and the structural equations simplify-, m

do'i =

Qa A Wai

a=1

dwij = E w1a A ula j - K* ai A o'j. a=1

In the middle of the 19th century. Riemann gave the complete local description of these spaces.

Theorem 51 (Riemann, 1854). Each space form is locally isometric to the spherical space S'", the euclidian space R"', or the hyperbolic space 1.tm.

5. Curves and Surfaces in Euclidean 3-Space

198

Exercises 1. Let y(s) be a curve in its natural parametrization, and let p(t) be this same curve in an arbitrary parametrization, which hence is related to 7 via Ea(t) = (s(t))a) Prove the following formula for the second derivative of p: d2µ(t) _ d27 ds 2 dy d2s = (t) (s)2 h + s t µ(t) = dt2 182 + ds dt2 b) Deduce from this the formula for the curvature in a general parametrization:

(dt

.

mo(t) = IIi(t) X µ(011 IIi (t)113

c) Prove that the torsion can be written in the Frenet frame as det (7', 7", -y"')

r(t)

(Fx h', h) = det (t, h, h') =

K(t)2

d) Derive in an analogous way to a) a formula for -ii (t), and conclude that the torsion is given in an arbitrary parametrization by det (µ(t), µ(t), i (t))

r(t) =

IIi (t) x A(t)112

2. Let C C R3 be a curve all of whose tangents pass through one and the same point. Prove that C is part of a straight line. 3. Let C C 1123 be a curve all of whose tangents are parallel to one and the same straight line. Prove that C is part of a straight line.

4. Show that the tractrix (Example 11) is the curve passing through the point (1.0) on the horizontal axis with the property that the length of the segment of the tangent line from any point on the curve to the vertical axis is constant. The following intuitive interpretation gave it its German name "Schleppkurve"(tow curve): It is the path that a dog pulling on a leash to the west is constrained to take when his master walks along a north-south path. 5. Compute the curvature and the torsion of the following curves:

a) y(t) = (t. a(e'/° + e-t/a)/2, 0) (catenary);

199

Exercises

b) -y(t) = (a cost, a sin t, b t) (helix);

c) y(t) = (a(t

- sint), a(l - cost), bt);

d) y(t) = (t, t2, t3).

6. Prove that y(t) = (at, b t2, t3) with a, b > 0 is a slope line if and only if 2b2 = 3a.

7. Prove that a C3-curve C is a helix if and only if its curvature r, > 0 and its torsion r are constant. 8. Determine all plane curves C C R2 satisfying

a) K(s) = const; b) c(s) = 1 c) K(s) =

9. The Darboux vector of a curve with non-vanishing curvature is the vecto Prove that the Frenet formulas can be written in the following C T:= r form: ds

= d x t,

ds

=

j -X hh,

= d *X 6.

ds

10. Consider the spherical curve y*(s) := t(s)

S2 C iR3 and suppose that the curvature K(s) of the original curve y(s) does not vanish. Prove that 'y' (s) is a regular curve, and compute its curvature K' and its torsion r'. Which curve y*(s) arises in the case of the helix y(s)? :

[0, L]

11. A slope line with slope angle a lies on a sphere of radius R if and only if its curvature K(s) and its arc length s satisfy the relation 2 S

tan2 a + K2 (S)

_ R2

12. Let y(s) : [0, L] -> R2 = C be a closed plane curve, and let p be a point in the plane not belonging to the curve. Prove that the integral L t(s)

1j

21ri

-Y(s) - P

ds

is an integer. This is called the index or winding number of the curve y(s) around the point p.

5. Curves and Surfaces in Euclidean 3-Space

200

13 (Jacobi, 1842). Let -y C R3 be a closed curve with non-vanishing curvature, and suppose that the curve of principal normal vectors h is a simple closed curve in the sphere S2. Prove that h divides the sphere into two parts of equal area. 14. Classify all surfaces of revolution with constant Gaussian curvature using the results from Examples 3 and 11. Altogether there are 9 geometric types. Which of these can be compact manifolds without boundary? 15. Let F : U -* 1R3 be a parametrized surface such that the tangent vectors

r9/au and a/av are orthogonal at each point (u,v) E U. Set

E_

\ 49u, au. >

,

C= \

and choose the orthonormal frame _ 1 8

au' e2 _

e

a

8v49

' Ov

1

a

av

a) In the dual frame al = /Edu, 02 = /dv, the connection form w12 is given by

wl2 =

1

aV G

vE - av

al +

&

b) The Gaussian curvature is equal to

K = - EG ava

a (8,/-G/8u)j

+OU

E

c) As an application, determine the Gaussian curvature of a surface of revolution whose generating curve is not necessarily parametrized by arc length.

16. Let 1112 C 11 P3 be a surface with non-vanishing Gaussian curvature. Prove:

a) There exist two orthogonal vector fields V and W with lengths IIVII = 1 and IIWII = IHI whose flows are volume-preserving, Gy(dM2) = Gw(dM2) = 0. b) The integral curves of the vector field W/IH) are geodesic lines.

c) Let B : M2 - R be the function measuring the geodesic curvature of the integral curve of the vector field V at a point. Then W(H) = H2 B. d) The following commutator formula holds: [W/IHI, V) = B V.

Exercises

201

17. Discuss the geometric properties of the helicoid and the catenoid, and study the map transforming the one into the other.

18. Enneper's minimal surface is parametrized by (z = a + ib) F(a, b) = (Re(3z - z3), Re i(3z + z3), Re(-3z2)) . Compute its Gaussian curvature. Prove, moreover, that its curvature lines (i. e., the curves on the surface obtained by fixing one of the two parameters a or b) all lie in a plane through the y- and z-axis and have polynomial arc length. Determine the equation for the tangent plane to a point x E F and transform it into (p - q, x) = (p, p) - (q, q) This is the equation of a plane with respect to which the two points p and q are mirror images of one another. The curves p(a), q(b) are parabolas in the perpendicular planes x = 0, y = 0, and the summit of each of them coincides with the focus of the other (so-called focal parabolas). Illustrate these properties using Maple or Mathematica. 19. The minimal surface studied by Bour (1862) arises from the Weierstrass

representation by inserting f (z) = czm, g(z) = z. Prove that it can be mapped to a surface of revolution in a length-preserving way.

20. Prove the following properties of the modular surface of a holomorphic function f : U -* C:

a) The modular surface is hyperbolically curved in a sufficiently small neighborhood of a finite pole of f. b) Let zo be a finite zero of f . If this zero is multiple, the modular surface is elliptically curved in a sufficiently small neighborhood of zo. If the zero is simple, every sufficiently small neighborhood of zo decomposes into 2k - 2 sector-like pieces, in which the modular surface is alternatingly

curved elliptically and hyperbolically. Here k is the exponent of the non-vanishing term succeeding the summand a, (z - zo) in the series expansion of f. In the next two exercises it will be helpful to use the position functions pi from the proof of the Minkowski-Steiner theorem.

21. Let M2 be a surface with normal vector e3. Prove that the 1-form = (x, de3) defined on the surface is closed, and conclude that, for a 77 1

curve ry C M2, the relation

j(xde3) = 0

202

5. Curves and Surfaces in Euclidean 3-Space

holds if one of the following two conditions is satisfied: a) The curve y is the boundary of a two-dimensional submanifold G2 C M2. b) 111 is simply connected.

22. Let A12 be a compact surface without boundary. Prove that there exists a point ro E A!2 with the following properties:

a) The normal vector e3 is parallel to the position vector rO at the point ro. b) The Gaussian curvature is positive at xo, G(xo) > 0.

Hint: The function f : M2 -+ R, f (x) = (x, x), has to have a critical point on Ate.

23. Let the group G = SL(2,R) act on the hyperbolic plane ?{2 by _ a b .z _ az+b

cd

6F

cz+d'

Verify that the image point g z actually does belong to ?{2, and that superposition of two of these transformations corresponds to matrix multiplication in G. Show. moreover, that each g E G leaves the metric invariant; hence G is an 3-dimensional group of isometries from N2 onto itself.

24. Let AI'; = 1R3 and denote by DyX the directional derivative of the R3 in the direction of the vector X. Prove that VXY := DXY+2X xY vector-valued function X : R3

defines a covariant derivative having all the properties (1)-(4) from Theorem 42. but violating property (5).

25. Consider on the set {(x,y) E R2 : -7r/2 < y < 7r/2} the pseudoRiemannian metric

1 _ dx2-dye x'2

- y'2

COS2

x'

a) Prove that E _2(y) and P =2(y) are first integrals of the cos

geodesic flow, satisfying, in addition, the inequality p2 - E > 0. b) Discuss the geodesic lines on M2. To do so, assume that y is a function of r and integrate the resulting ordinary differential equation.

c) On Al" there exist points that cannot be joined by a geodesic line.

203

Exercises

26. Prove that every three-dimensional Einstein space is a space of constant curvature.

27. Let M' be a non-flat Einstein space (for example, S"'). Show that MI x M"' with the product metric is an Einstein space, but not a space of constant curvature.

28. It is well-known that a symmetric bilinear form h.(x, y) on a vector space is completely determined by its quadratic form q(x) := h(x, x) (via the polarization formula: 2h(x, y) = q(x + y) - q(x) - q(y)). Prove, in a similar way, that the Riemannian curvature tensor 1 (U. V, W1, W2) is completely determined by the quadratic form K(U, V) := R(U, V, V, U) corresponding to the sectional curvature. 29. Prove that a four-dimensional Riemannian manifold is an Einstein space if and only if for every 2-plane E2 C TM4 and its orthogonal complement (E2)1 the corresponding sectional curvatures coincide, K(E2) = K((E2)1).

30. Because of its symmetry properties, the curvature tensor of a pseudoRiemannian manifold can be interpreted as a transformation R : A2(M'') A2 (M7°) on 2-forms, n 1

R(ai Aaj) = 2 > Ro3i . a, Aa,3. Q,:3=1

In dimension m = 4, this gives rise to an endomorphism R : A2(A14) A2(M4) A2(M4). On the other hand, the Hodge operator * : A2(A14) acts on A2(M4), and its square depends only on the index k of the metric (see

Theorem 5, Chapter 1): ** = (-1)k. In the cases k = 0 (positive definite metric) and k = 2 (neutral metric), the Hodge operator decomposes the real bundle A2(M4) into the corresponding eigenspaces A2 (M4) (see Exercise 8, Chapter 1). Prove that the block representation of the curvature tensor.

R _

R++ R-+

R+- R--

with respect to this decomposition of A2(M4) has the following properties:

a) 7Z is symmetric, i.e., R++ = R++, R__ = R__ and R+_ = R_+.

b) The traces of R++ and R__ coincide, tr(R++) = tr(R_-) = -r/12. c) M4 is an Einstein space if and only if R+_ vanishes.

5. Curves and Surfaces in Euclidean 3-Space

204

Literature: Th. Friedrich, Self-duality of Riemannian manifolds and connections, in: Riemannian geometry and instantons, Teubner-Verlag, Leipzig, 1981, 56-104.

The Einstein equation in the general theory of relativity combines the geometric curvature quantities of a four-dimensional pseudo-Riemannian manifold of signature (1, 3) with its physical properties encoded in the energymomentum tensor T,

Ric-

rcT.

Here Ric. g. r are the Ricci tensor, the metric, and the scalar curvature; K is a constant depending on the chosen system of units. Already for the vacuum, T = 0, there are non-trivial, physically very interesting solutions of this equation, among others the Schwarzschild metric to be discussed now. Obviously, a vacuum solution of the Einstein equation has to be an Einstein space with vanishing Ricci tensor in the sense of the definition given before. 31 (Schwarzschild metric). On a spherically symmetric and static space-time manifold M4 (this is a pseudo-Riemannian manifold of signature (1, 3) with

isometry group SO(3, R) and a distinguished time direction), it is always possible to introduce coordinates from R x R+ x S2 with respect to which the metric can be written as g = e2a(r) dt2 - [e 2b(r) dr2 + r2 (d92 + sin2 O dV2) J .

Here, the functions a(r) and b(r) asymptotically tend to zero for r - co (the metric is "asymptotically flat"). We introduce the following basis of 1-forms:

ao = ea dt, Cl = eb dr,

a2 = rdO, 03 = r sin 9 dcp .

a) Check that the metric satisfies g = 0,02 - 0i - 02 - a3 . b) Compute the forms dai and show, using the first structural equation, that the connection forms are given by wo, = -a'e-bcO,

e-b

WO2 = W03 = 0,

a-b

W12 = -r 0`2, w13 = r- a3, w23 =

cot 9

r

a3-

c) We introduce the notation lid := 2 E.4 Rtasia0 A0,3. (f is the so-called curvature form). Show, using the second structural equation, that

110 = e-2b(a'b'-a"-a2)aoAal, f20 = -a'er"aoAa2, -a'er6

03O =

ao A 03,

5231 = b'erbal A a3,

1121 = bier

6a1

A02,

X32 = '-'-'012 A a3,

Exercises

205

and compute from these formulas the components R kl of the curvature tensor in this basis. d) Compute the components of the Einstein tensor G := Ric - 1g ,r: ) , Gi l = ,1-s _e-2b(77 Goo= I e-2b (13 _ 2b F 7-

-

+),

G22 = G33 = -e-2b (a'2 - a'N + a-+

=d) r

e) Solve the vacuum equation by means of this Ansatz. The result is (M is a constant of integration) r

g = 11-

1

2M dt2 -

r

J

r

11I

2M

r

1_1

dr2 J

- r2 (d92 + sin2 0 d 2)

32. Restrict the Schwarzschild metric g to the two-dimensional submanifold

defined by 0 = x/2 and t = const. Prove that this yields the metric of a paraboloid of revolution (a "lying" parabola!) with the equation z2 = 8M(r - 2M) (see the picture). 33. Light moves along geodesic lines whose tangent vectors have length zero. Making use of the fact that the equations defining a geodesic are precisely

the Euler-Lagrange equations of the length function G = ta gjjx'ia, prove the following assertions:

a) A particle moving at t = 0 in the equatorial plane 0 = x/2 stays there forever.

b) The quantities L := r2cp and E := t(1-2M/r) are first integrals. Moreover, the second Kepler law holds: The orbit ray covers equal areas in equal times.

5. Curves and Surfaces in Euclidean 3-Space

206

c) Set r = r(W) and derive the equation describing the motion of light rays.

Result: It is reasonable to set u := 1/r: u" + u = 3Mu2. d) Solve this differential equation approximately up to second order in V. Result:

uo =

1

sin V +

[1

+

cos 2W

.

Interpret the constant of integration 2 0 ro as 3the scattering length. Which asymptotic value arises for cp if r tends to oo and sp is supposed to stay small? Twice this value, denoted by 5 in the picture, is the relativistic deviation of light in the gravitational field of a very large miss, which can be described by the Schwarzschild metric.

Chapter 6

Lie Groups and Homogeneous Spaces

6.1. Lie Groups and Lie Algebras In the preceding chapter Noether's theorem showed that symmetry considerations simplify the study of geometric problems, and sometimes it is only by symmetry considerations that a solution is possible at all. In fact, beginning in the 1870s, the conviction grew that the basic principle organizing geometry ought to be the study of its symmetry groups. In his inaugural lecture at the University of Erlangen, which later became known as the "Erlanger Programm", Felix Klein said, in 1872, "Es ist eine Mannigfaltigkeit and in derselben eine Transformationsgruppe gegeben; man soil die der Mannigfaltigkeit angehorigen Gebilde hinsichtlich solcher Eigenschaften un-

tersuchen, die durch die Transformationen der Gruppe nicht geandert werdenl.

One has to distinguish whether the groups under consideration are discrete (for example permutation groups) or continuous (for example, oneparameter groups of isometries). The latter were systematically introduced by the Norwegian mathematician Sophus Lie (1842-1899). which is why they bear his name today.

1 "Let a manifold and a transformation group in it be given; the objects belonging to the manifold ought to be studied with respect to those properties which are not changed by the transformations of the group."-quoted from F. Klein, Des Erlanger Programm, Ostwalds Klassiker der exakten Wissenschaften, Band 253. Verlag H. Deutsch, Frankfurt a. M., 1995, p. 34. 207

6. Lie Groups and Homogeneous Spaces

208

The fundamental idea of a Lie group is a very simple one. It ought to be a group which is at the same time a manifold, and hence allows a differential calculus. Moreover, the manifold structure has to be compatible with the group structure, i. e., the product is a differentiable map.

Definition 1. A Lie group is a group G which, at the same time, is a differentiable manifold such that the map is (infinitely often) differentiable.

Remark. Obviously, the last condition is equivalent to requiring that the product map and the inversion (9, h) -. g . h,

g'-' g-1 ,

be differentiable maps.

Example 1. Every finite-dimensional vector space V is an abelian Lie group whose group product is exactly the addition of vectors (v, w) H v + w.

Example 2. The unit circle S' = {z E C : IzI = 1} is an abelian Lie group with the usual multiplication of complex numbers as product.

Example 3. Most Lie groups can be realized as matrix groups. The set of all invertible matrices with entries in K = R or C is an open subset of IK"z and hence a manifold. Endowed with matrix multiplication as product, it forms a Lie group, the general linear group

GL(n,K) := {A E .M"(K) : det A 540} . More generally, GL(V) is meant to denote the group of invertible endomorphisms of the vector space V.

Example 4. The vector space K" and the general linear group GL(n, K) combine to form a new Lie group, the affine group Aff (K") := GL(n, K) x IIS"

with multiplication

(A, v) (B, w) := (AB, Aw + v). This product rule arises in a natural way by defining an action of the affine group on the vector space K" through

Aff(K") x K' V. (A, v)x := Ax + v. and then applying the transformations determined by (B, w) and (A, v) on a vector x one after the other. Each Lie group G acts on itself by means of the left and the right translation with a fixed element g E G,

L9, R9 : G -. G, L9(h) = g h and R9(h) = h g.

6.1. Lie Groups and Lie Algebras

209

The corresponding differentials are the following maps in the tangent bundle of G: Th9G. (dLg)h: ThG - TghG, (dR9)h : ThG To avoid double indices, in this chapter we will use the notation (df)h instead of f.,h for the differential of a map f.

Definition 2. A vector field X on G is called left-invariant (or rightinvariant, respectively) if it is transformed into itself by dL9 (or dR9, i.e., dL9X = X. At a point h E G, this means

(dLg)hX(h) = X(g - h). Since left translation is obviously a diffeomorphism of G, Theorem 35 in §3.9 can be applied to yield, for the commutator of two left-invariant vector fields, the formula

dLg[X,Y] = [dL9X,dLgy] = [X,Y1. This property, together with the fact that vector fields satisfy the Jacobi identity (Theorem 34, §3.9), endows the vector space of left-invariant vector fields with the structure of a Lie algebra, the Lie algebra g of the Lie group G.

Theorem 1. The vector space of left-invariant vector fields on a Lie group G is canonically isomorphic to the tangent space at the neutral element,

g='TeG. Proof. With each left-invariant vector field X, we associate its evaluation at the neutral element e, X - X(e) =: X E TOG. Conversely, every element X E TeG determines a vector field Xx on G by setting

Mx(g) := (dL9)e(X) This satisfies the relation

Xx(gh) = (dLgh)e(X) = (dLg)h(dLh)e(X) = (dLg)hXX(h)

0

hence Xx is left-invariant.

Remark. Because of this fact, we will no longer distinguish between the Lie algebra of left-invariant vector fields and the tangent space to the group

at the neutral element. Its elements will be denoted by upper case Latin letters X, Y, ... E 9. Choosing a basis X1, ..., Xr of 9, we can again write their commutators as linear combinations of the basis elements, r

[Xi,Xj] = k=1

&Xk-

6. Lie Groups and Homogeneous Spaces

210

The antisymmetry of the commutator and the Jacobi identity imply that the constants C have to satisfy the relations

Cij - - Cii '

CijCk,n + Cj' "ki + CrniCkj = 0.

Following E. Cartan, the numbers C are called the structure constants of the Lie group G, since Cartan's structural equations are simple to formulate in their terms. To see this, we agree to call a differential form w on G left-invariant if it satisfies the condition L*9w = w.

Following the argument in the case of vector fields, it is easy to see that the r-dimensional vector space g* of left-invariant 1-forms is canonically isomorphic to Te G. Now let al, ...,o-, be the basis of g* dual to X1. ..., Xr. Theorem 2 (Maurer-Cartan Equations). Let C be the structure constants of a Lie group G with respect to the basis X1, ... , Xr of its Lie algebra g. Then the exterior derivatives of the forms in the dual basis al, ... , or of g* are given by

do, _ -r k of Aak. j
Proof. From Theorem 33, §3.9, we know the general formula for the differential of a 1-form oi:

do (Xj, Xk) = Xj (0t(Xk)) - Xk(ai(Xj)) - ai([Xj, Xk)) Since the forms ai are dual to the basis vectors Xj, the first two terms on the right-hand side vanish. Now

dn'i(Xj,Xk) = -ai([Xj,Xk]) = -ai(E,nC7kXm) = -Cjik, which is precisely the assertion.

We introduce a 1-form e : T(G) -+ g defined by the formula

e(t) := dL9-1(t) for every tangent vector t E T9(G) at the element g E G. The form e shifts the tangent vectors of the Lie group via left translations to the space Te(G) = g. It is the so-called Maurer-Cartan form of the Lie group. In general, if el, e2 are two 1-forms with values in the Lie algebra g, we define

its "exterior" product as a 2-form with values in the Lie algebra by the formula

[ei,e21(tl,t2) [el(tl),e2(t2)] - [el(t2),e2(tl)l In particular, [e, el(tl, t2) = 2 - [9(tl ), e(t2)],

6.1. Lie Groups and Lie Algebras

211

and the Maurer-Cartan equations can be formulated equivalently as

0. Example 5. The special linear group in dimension two, SL(2, R) = {A E M2 (R) : det A = 1 } , is a 3-dimensional Lie group with Lie algebra (see Exercise 5, Chapter 3) s1(2, R) = {X E M2(R) : tr A = 0 } . The matrices

H = 10

011' E

[0 01' F

[1 0,

form a basis of sl(2, lR) and satisfy the comniutator relations

[H, E] = 2E.

[E, F] = H.

[H, F] = -2F,

If we denote the forms of the dual basis of sl(2, I8)` by ax, 0E, and aF, the Maurer-Cartan structural equations read as

dvy = - CE A QF, dOE = -2 aH A O'E,

doF = 2 otl A aF .

Example 6. The Lie group GL(n, K) is an open subset of M. (K). Its Lie algebra is therefore

gl(n,K) '=' Te GL(n, K) = M,. (K) . Theorem 3. Let X be a left-invariant vector field on G and denote by 4x (t) the maximal integral curve of X such that 4x (0) = e. Then

(1) 4x(t) is a group homomorphism. 4x(t1+t2) = 4'x(tt) ° 4>x(t2) (2) the vector field X is complete, i. e.. 4b,y(1) is defined on all of R.

Proof. Let 4bx(t) be defined on the interval to < t < tb. We start by showing the first assertion for two values tl, t2 such that they together with their sum t t + t2 belong to the domain of definition of 4x (t). To do so, fix t 1, set. g 4,x (t 1), and consider the curve

z1(s) = g 4x(s) on the interval (ta, tb) as well as the integral curve obtained by shifting the argument by tt,

zit (s) = Ix(tt + s) on the interval (t,i - tt, tb - tt) . From their derivatives, dzi

,r

fd4x

r /-'- , \

1

v/

.

,

".

d and

dill ds

__

d4bx(ti +s) ds

= X (tx(tt + s)) = X (zlt (s)),

v/ ,

%%

6. Lie Groups and Homogeneous Spaces

212

we conclude that q and ill both are integral curves of X with equal initial conditions.

fi(0) = 'x(ti) = g = 9(0) Hence'i and r11 have to coincide on (ta, tb) fl (ta - t1, tb - t1). One easily checks that t2 has to belong to this intersection, i. e..

' x(tl+t2) =

0 o4x(t2)

Now we show that tb = 00; to prove ta = -oo one proceeds analogously. Suppose that tb is finite. Then,

r)(s) _ (px (2) x (s -

2

)

is a curve defined on [ta + tb/2, 3 tb/2)) with tangent vector

ds =

2) = X (sx() 4x(s -

2

= X(gl(s)).

Hence ()(.s) is an integral curve of X defined on [ta + tb/2, 3 tb/2) such that q(s) = 4 x (s) for s E (ta, tb) fl [ta + tb/2, 3 tb/2). Thus the maximal integral curve has to be defined at least on 3 tb/2, a contradiction to the maximality of tb.

This theorem enables us to define the exponential map for Lie groups.

Definition 3. The exponential map exp : g -. G is the evaluation at t = 1 of the integral curve 4 x of X E g satisfying 4x (0) = e, exp(X)

4x(l)

Theorem 4. The exponential exp : g -y G is a smooth map with the following properties: (1) exp(O) = e;

(2) exp(-X) = [exp(X)]-1; (3) exp ((s + t)X) = exp(sX) exp(tX). In addition, it is a diffeomorphism from an open neighborhood V of zero in g onto an open neighborhood of the neutral element in G.

Proof. For the smoothness claim, we have to refer to the fact that the solution of a system of differential equations depends smoothly on the initial conditions. Under scalar multiplication, the flow transforms as

Op (t) = 4Dx(A-t), since

d4x(At) = a X('Px(,\t)) dt

6.1. Lie Groups and Lie Algebras

213

together with'x(A 0) = e implies that cx(A t) is an integral curve of A X. Restricting the exponential map to a straight line through the origin yields a one-parameter subgroup of G. This proves property (3):

exp ((s+t)X) = 1D(s+t)x(1) = 4x(s+t) = 4tx(1) = exp(tX) exp(tX). Property (2) is an immediate consequence of (3). and property (1) is trivial. The final assertion follows from the inverse function theorem if we can show that the differential of the exponential map at the origin,

d(exp)o: Tog=9- TG=9. is invertible. But this is just the identity map: Consider the curve -Y(s) _ s X in g with tangent vector 7(0) = X. Then d(exp)o(s.X) =

dexp(sX) 18=o

_ dqbx(s) = d48x(1) ds I3=0 ds 8=0 = X(e).

ds Since we already know that the evaluation of a left-invariant vector field at I

the neutral element e is an isomorphism, the proof is completed.

It turns out that all one-parameter subgroups of a Lie group G have the form exp(tX) for some X E 9.

Theorem 5. Every continuous group homomorphism ' : R

G has the

form

11(t) = exp(tX) for a certain X E 9. In particular, every continuous group homomorphism 1R G is smooth.

Proof. Let 41

:

G be a continuous group homomorphism, and let

1R

V C g be a neighborhood of the origin so small that exp : g - G is a diffeomorphism on 2V. Let 11' = exp(V) be the image of V in G. Since %F is continuous, there exists an e > 0 such that 'P([-e, e]) C W. In particular,

fle) has to have a pre-image X E V under the exponential map, %P(e) _ exp(X). The smooth group homomorphism

f:

f(t) = exp (ix).

G,

coincides with 'F(e) at the point

.

Consider the set

K = {tEIR: 'F(t)= f(t)}. This is a closed subgroup of IR, since ' and f are continuous group homomorphisms. Therefore, K is equal either to R or to a discrete subgroup of it.

6. Lie Groups and Homogeneous Spaces

214

The latter case will be excluded, implying the equality of f and i. Assume for a contradiction that K is generated by a number a > 0,

K = In-a: nEZ}. As c already belongs to K, a has to be less than e, and '+(a) belongs to W. «'e show a/2 E K to obtain the desired contradiction. To do so. start from the Ansatz

f (a/2)2 = f(a) = 'I'(a) = 'I' (a/2)2 . Then, applying the inverse of the exponential map,

2exp-' (f (a/2)) = 2exp 1('' (a/2)) and noting that'P(a/2) belongs to IV, we conclude that 2exp-' (1P (a/2)) E 2V. On 2V. the exponential map is a diffeomorphism, and hence

f (a/2) = IP (a/2) .

0

This means that a/2 E K.

Definition 4. Let G1, C2 be two Lie groups, and let 0 : G1 -. C2 be a group homomorphism. We define the differential of io as a map between the Lie algebras of left-invariant vector fields on GI and G2 as follows: Consider

for X E g i the one-parameter subgroup exp(tX) E G1. By assumption, its image curve '(exp(tX)) is a one-parameter subgroup of G2, which by Theorem 5 has to have the form exp(tY) for some Y E 92. Now we set

y.(X) := Y.

u' : 91 - 92,

This definition immediately implies that the following diagram commutes: 92

jex C2

Identifying 91 - TeG1 as well as 92 ^_' TeG2, we see that this definition is compatible with the usual one of the differential at the neutral element, di'e : TeG1 -+ TE.G2. More precisely, the diagram extended by d,', remains commutative: v.

91

exp

G1

6.2. Closed Subgroups and Homogeneous Spaces

215

To see this, take any left-invariant vector field X E gl and compute _ dexp(tO.(X)) _ dV,(exp(tX)) _ dt _ .(X)(e) dt dVe(X (e)) I

=o

t=o

A further important property of the differential V. is its compatibility with the commutator.

Theorem 6. The differential of a homomorphism between Lie groups, GI -+ G2. is a homomorphism of Lie algebras, i. e..

:

14'.[X.Y] = [?P.X,V'.YJ-

Proof. The commutativity of the diagram just discussed is equivalent to the following identity, holding for every left-invariant vector field X E g:

iP.(X)(e) = dl/'e(X(e)). From this, we deduce the analogous identity holding at each point g E GI, d,;"9(X (g)) = dt''g(dLg)e(X (e)) = d(it' o L9)e(X(e)) = (dLg9))ed ke(X (e)) _ (dL, ls))e (X) (e) = w'.(X)(vI(g)). This means that the vector fields X and ,b.X are ?P-related in the sense of the remark following Theorem 35, §3.9. In particular, this remark implies the assertion.

6.2. Closed Subgroups and Homogeneous Spaces The exponential map of Lie groups does not satisfy the functional equation familiar from calculus; it turns out that a second order correction has to be made.

Lemma 1. Let G be a Lie group and let exp : g G be the exponential map. Then, for two arbitrary left-invariant vector fields X, Y E g, the following identity holds:

exp(tX) exp(tY) = exp (t(X + Y) + 0(t2)) Here 0(t2) indicates that 0(t2)/t2 stays bounded for t - 0.

Proof. Choose a neighborhood V C g, 0 E V. such that exp : V - W' C G is a chart around the neutral element e E G, and also choose a basis XI, ... , Xr of g. By means of (XI , ... , xr) _ exp(x1Xi +... + xrXr) E G, we introduce coordinates in W. Let U C W be so small that U U C W. (By " " we denote the product in the group G.) On U x U C G x G. we then have the coordinates

y' = x'opri, z' = xtopre

6. Lie Groups and Homogeneous Spaces

216

and, because U U C W, we can restrict the group multiplication to obtain

a mapp:UxU-+W. Let i

i

r

1

1

r

be the coordinate representation of p. Then

f`(0, ...,0,z1, ...,zr) = z', f'(yl, ...,yr,0, ...,0) = y', and hence f' has the Taylor expansion i

ft(y..) = f`(0,0)+

k(0,0)yk+

i

ayl(0,0)zl+R,

where the remainder R consists of second order terms. Inserting tY = y'Xi. tZ = z'Xi into the Taylor expansion, we now obtain

exp(tY + tZ + R) = exp(tY) exp(tZ), and the term R tends to zero like O(t2).

O

The following example shows that not every continuous subgroup of a Lie group has to be a Lie group itself.

Example 7. Let T2 = Sl x S' be the two-dimensional torus, parametrized by (t. s) E [0, 1] x [0, 1]. For every real number q, the formula yq : R T2, yq(t) = (t, q t) defines a curve in T2, which is isomorphic to the additive group R of real numbers. We want to show that this curve is dense in T2 for irrational q. and hence provides an example of a connected, non-closed oneparameter subgroup of a Lie group. Let (to, so) E [0,1]2 be an arbitrarily given point on T2, which we want to approximate by a sequence of points in yq. For each integer n E Z, we have

,y(to+n) = (to+n,qto+qn), and, as an angle coordinate in S1, to + n is equivalent to to. Similarly, qto + qn + m is equivalent to qto+qn for every integer m; hence it corresponds to the same point on yq and on T2. If we are able to show now that qZ + Z

is dense in R. then we could conclude the existence of a sequence of pairs (n;, m,) E Z x Z for which

limgni+mi = so-qto.

i-00

But then q(to + ni) + mi converges to so for i - oc, and lim ryq(to + nt) = 'Y(to, so), as claimed. Thus it remains to prove the following lemma.

Lemma 2. For every irrational number q, the set Z + qZ is dense in R.

6.2. Closed Subgroups and Homogeneous Spaces

217

Proof. Like the set Z+qZ, its closure is a subgroup of R. The group Z+qZ cannot be cyclic; if it were, it would be generated by an element a > 0, and there would exist integers k and I such that

I=

q = Va.

But this would imply q = Ilk E Q. Hence we arrive at a contradiction. As the closed subgroups of IR are precisely the cyclic groups and R itself, only this last possibility remains for Z + qZ. In Example 4, §3.1. we encountered a parametrization of a torus of revolution

by means of S' x S'. The following pictures show the trace of a curve yy in this parametrization for two close rational and irrational values of q, respectively. In Example 8, §7.4, we show that the motion of a spherical pendulum can be parametrized precisely by such a curve on a torus.

The main objective of this section is to prove that a closed subgroup H of a Lie group G is a submanifold of it. Consequently, it is itself a Lie group, and the quotient space C/H carries the structure of a manifold with a smooth G-action. As a preparation, we need a few technical lemmas. Let II - II be any norm on the vector space g. Lemma 3. Let H be a closed subgroup of G. and let Xn # 0 form a sequence converging to zero in 9 such that exp(X,,) belongs to H and Xn/IIXnII tends to an element X E g. Then

exp(tX) E H for all t E R. Proof. For a fixed t > 0 we define a sequence of natural numbers m by

mn := max{kEN: Then the following estimate holds:

mnIIXnII < t < (mn + 1)IIXnII = mnIIXnII + IIXnII But on the right-hand side, the sequence IIXnII tends to zero, and hence

lim mnIIXnII = t. n-or,

6. Lie Groups and Homogeneous Spaces

218

This implies

limn rnXn =

hn-oo m m HIX,,II

IlXnll

= tX,

and, since the exponential map is continuous,

exp(t X).

Iim

n-oo Each term in the sequence exp(m,,Xn) = [exp(X,,)]"'" is an element of H. As H is closed by assumption, the limit of this sequence, exp(t X), has to lie in H, too. The proof for the case t < 0 proceeds along the same lines.

Lemma 4. For every closed subgroup H C G, the set

b :_ {XEg: exp(tX)EHforalltElR} is a linear subspace of g.

Proof. For any vector X E h, its scalar multiples a X also belong to h. It thus suffices to show that h is closed under addition. To this end, let X, Y be elements of h, and suppose that X + Y 36 0. In any case, the product exp(tX) exp(tY) belongs to the subgroup H, and for sufficiently small t we have, by Lemma 1,

exp(tX) exp(tY) = exp(t(X +Y) + 0(t2)). Then Z(t) := O(t2)/t apparently converges to zero for t - 0, and we can rewrite the preceding equation as

exp(tX) exp(tY) = exp (t(X +Y + Z(t))) E H. Choose a sequence of positive numbers that converges to zero, t - 0, and define Xn := tn(X + Y + Z(tn)). Each term exp(XX) lies in H, and --:in X+Y = lim X+Y+Z(t,) lim n-- IIX + Y + Z(t,a)[I = Fix -+Y11 11X-11 Obviously, we have Xn # 0 and Xn -' 0. Thus, Lemma 3 applies, and we can conclude that for every t E R

exp(t

IIX+YII

is an element of H. Hence X + Y belongs to Fj. Suppose that H is, in addition, a closed subgroup of G, and let h be defined as in the preceding lemma. We then choose any linear complement h' of h in 9,

9 = h+h'. Lemma 5. There exists a neighborhood V' C 4' of 0 such that, for every X' 34 0 in V', the element exp(X') does not belong to H.

6.2. Closed Subgroups and Homogeneous Spaces

219

Proof. If the assertion were false, there would exist a sequence X;, E h' converging to zero and satisfying exp(X,,) E H. Now consider the compact set K :_ {X' E h': 1 < JJX'II < 2} and choose natural numbers p,, such that p ,,X,, E K. Since K is compact, we may assume that the sequence p,X;, converges to some 0 9& X' E K. Again, [exp(X;,)]P is an element of H and lim

pnX' 11p-Xnii

_

X' 11X'6

Then Lemma 3 implies that X'/IIX'II E h, contradicting 0 0 X' E '.

0

Now we can turn to the main theorem of this section.

Theorem 7. Let G be a Lie group, and let H be a closed subgroup. Then: (1) H is a submanifold of G and thus itself a Lie group. (2) There exists precisely one differential structure on G/H such that (a) the projection 7r : G - G/H is smooth, (b) for every p E G/H there exist a neighborhood Wp C G/H of p and a smooth map w : Wp G such that 7r o p = Idw,,, (c) the action of G on G/H defined by (g. kH) gkH is smooth. Proof. It obviously suffices to show that there exists a neighborhood W C G of e for which H n W is a submanifold (left translation is a diffeomorphism of G). As before, ddecompose the Lie algebra into g = 1) + h' and consider the map corresponding to this decomposition, 4D :

g = h + 4' ---i G, 4(X + X') = exp(X) exp(X').

In h' we choose a neighborhood V' C h' as in Lemma 5. as well as a subset V C h so small that the exponential map still is a diffeomorphism on V +V'. The image W of V + V' under -ID is an open neighborhood of e E G, and

HnW= by the definition of b and Lemma 5. The set H fl W is thus parametrized by the chart (V, -D Iv+(o}), and hence a submanifold of G.

Now we turn to the proof of the second assertion. Let 7r : G - G/H denote the projection. We define a topology on G/H by the condition

A C G/H is open :a 7r-' (A) C G is open. It is called the quotient topology on G/H, and it is designed to render the map 7r continuous. Endowed with this topology, G/H is a Hausdorff space (see Duistermaat/Koik, Lemma 1.11.3). To verify the properties a manifold

6. Lie Groups and Homogeneous Spaces

220

has to satisfy, consider the distinguished point xo := e H E G/H together with the sets V, V' introduced in the first part of the proof. The map

' : V' - G/H, X'

a(exp X')

,

is continuous and maps V' onto an open neighborhood U of xo. Moreover, V, is injective, since v(X') = O(Y') implies the existence of an element h E H

such that exp(X') = exp(Y') h. Hence

h = exp(X') exp(-Y') = b(0 + (X' - Y')) . Thus h also belongs to the set W, which was defined as the image of V + V

under C Since we already proved H fl W = 4(V + {0}), this implies X' = Y'. In summary, the map 1/i : V' U is continuous and bijective. For an arbitrary point gH E G/H, consider the left translation by g E G on

G/H. L9 : G/H

G/H, kH

gkH

and introduce a chart around gH E G/H by L9(U),

U9H

09H: V' - UgH,

Z'gH := Lg o TV .

For two points gH and kH, the chart transition can be rewritten as follows, kH o y9H =

=

v-1

o Lk-1 o L9 0 V1 = exp-1 o(ir-I o 4-1 0 L9 0 a) o exp

exp-1 oLk-1h

oexp .

Therefore. as a superposition of smooth maps, the chart transition is also smooth. Hence we have proved that G/H is a differentiable manifold, the projection ;r : G - G/H is smooth, and C acts smoothly from the left on G/H. It remains to show (b). For the distinguished coset p = .ro = e H, define y; for each x in U =: Wp by

cp(x) = exp(Vi-1(x)) = 7r-1(x). For an arbitrary point p = gH one again uses the left translation L9.

0

Definition 5. The action of a Lie group G on a manifold Al is called transitive if, for two arbitrarily given points x and y in Al. the one can always be written as the image of the other under the action of G. i. e., there exists a g E G such that y = g x. An equivalent formulation of this requirement is to say that M consists of a single G-orbit, G x = Al. A manifold together with a transitive group action is also called a homogeneous space.

Obviously, the left translation on the quotient Al = G/H is a transitive group action, and thus G/H is a homogeneous space. Theorem 7 can be applied to show that some well-known matrix groups are Lie groups: The following groups apparently are closed subgroups of GL(n, K).

6.3. The Adjoint Representation

221

Example 8. The subgroup of GL(n, K) consisting of all matrices with determinant 1 is a Lie group, the special linear group,

SL(n,K) :_ {A E Mn(K) : det A = 1} .

Example 9. Let H

I =: h u, v E c } be the vector space of {[. Hamilton's quaternions with standard basis V

111

1

110

11'

oil' J Lof OJ, K = Lo i 01 and norm N(h) := uu + vv. The group of all quaternions with norm 1 is

E=

1=

LO

isomorphic to the Lie group

SU(2) := JA E GL(n, IC) : AAt = 12 and det(A) = 1). Example 10. The preceding example can be generalized as follows. The unitary group is embedded into the space of complex matrices as

U(n) :_ {A E

AAt = 1n}

.

The condition AAt = 1n immediately implies I det Al = 1, hence det A E Sl; the special unitary group is defined as the group of all unitary matrices A satisfying det A = 1:

SU(n) := {A E U(n) : det A = 1} . Example 11. The orthogonal group O(n, K) consists of the matrices A E M ,,(K) leaving the euclidean standard scalar product of Kn invariant,

(Ax, Ay) _ (x, y) Realizing the scalar product as (x, y) = xty, we see that this condition is equivalent to AAt = ln. Hence we obtain

O(n, K) = JA E Mn(K) : AA' = 1n} . Obviously, an orthogonal matrix has determinant + 1 or -1. The subgroup of all orthogonal matrices with determinant +1 is called the special orthogonal group SO(n, K),

SO(n,K) = {AEMn(K): AAt=lnand detA=1}.

6.3. The Adjoint Representation Definition 6. Let G be a Lie group with Lie algebra g, and let V be a finite-dimensional vector space.

6. Lie Groups and Homogeneous Spaces

222

(1) A representation of the Lie group C on V is a smooth group homomorphism e : G GL(V), i.e., a smooth map compatible with the group structure,

e(g h) = e(g) e(h) (2) A representation of the Lie algebm g on V is a homomorphism of Lie algebras, p : g - gl(V), i. e., a linear map compatible with the commutator,

e([X, Yl) = [e(X ), e(Y)] = e(X) e(Y) - e(Y), e(X) Sometimes, V is then also called a G-module or a g-module, respectively.

Example 12. The trivial representation of a Lie group G is the group homomorphism that maps every element g E G to the neutral element in GL(V): p(g) = 1V; the trivial representation of g associates the zero map with every element X, o(X) = 0v. Example 13. Matrix groups are defined by means of one of their representations, often called the defining representation. In fact, we introduced the groups GL(n, R), SL(n, R) and SO(n, R) in a way endowing them naturally with a representation on R". A simple example illustrates that these matrix groups and their Lie algebras have many more representations. The Lie algebra sl(2, R), for example, has representations in all dimensions: For every natural number n, define e : s((2, R) - gl(n + 1, R) by

g(H) = diag(n, n - 2, ..., -(n - 2), -n), 0

0 n

1

0

2

e(F) =

e(E) 0

0

n-1

0

nL

1

0

These matrices satisfy the commutator relations of sl(2, R), [e(H), e(E)1 = 2,o (E),

[e (H), e(F)l = -2e(F), [e(E), e(F)] = e(H), and hence form an (n + 1)-dimensional representation of sl(2, R). Properties

that cannot be expressed by the Lie bracket do not have to be preserved under a representation: For example, we have E2 = 0, but e(E)2 0 0. Nevertheless, the property that g(E) is a nilpotent matrix is preserved. There is also a a representation of the Lie group SL(2, R) corresponding to this representation of the Lie algebra; this will be the subject of Exercise 4.

Apart from left and right translation, there is a third remarkable action of a given Lie group G on itself, the so-called conjugation action,

ag : G - G, a9(h) := ghg-' = L9R9-, h.

6.3. The Adjoint Representation

223

It is smooth and satisfies ag(e) = e, and in contrast to left and right translation, it is far from being transitive. In the case G = GL(V), it decomposes the invertible matrices precisely into their similarity classes. In addition, the relation ag(e) = e implies that its differential at e is a map from g to g, d(ag)e: TG 5--- g ----+ TG 2--- 9,

which is obviously invertible, since d(Lg)e and d(Rg-i)e are invertible. We define the adjoint representation of G on g by Ad : G ---+ GL(g),

Ad(g) = d(ag)e E GL(g).

Before we verify that this actually is a representation, recall the definition of the center of a group. It consists of those elements which commute with all the others:

ZG = {gEG: gh=hgdhEG}. Theorem 8. The map Ad : G - GL(g) is a representation of G on the vector space g. The center ZG of G is contained in its kernel, ZG C ker Ad,

and equality holds if and only if G is connected.

Proof. First we check the homomorphism property:

Ad(gh) = d(LghRgh )e = d(LgLhRh-, Rg-i)e = d(agah)e = d(ag)ed(ah)e = Ad(g)Ad(h). Let z belong to the center Z. Then aZ = IdG, and hence Ad(z) = IdGL(9), i.e., z is in the kernel of Ad. Now suppose that G is connected and that

Ad(g) = Ide. Since ag : G - G is a group homomorphism, the map t ' -+ ag (exp tX) is a one-parameter subgroup for every X E g, and, by Theorem 5, there exists an element Y E g such that

exp(tY) = ag(exptX). Differentiating this equation with respect to t, we obtain Y(e) = dt (ag(exp tX )) I e=o = Ad(g) (X (e)) = X (e) , which proves X = Y. For the one-parameter group defined above this means

that exp(tX) = ag(exptX) for all t E R and X E 9. The exponential map is a local diffeomorphism g - G; hence ag = IdG on an open neighborhood W of e. For a connected

224

6. Lie Groups and Homogeneous Spaces

Lie group G this implies ay = IdG, since G can be represented as the union of all powers of W (with respect to the group products), 00

G = UW'. As a9 = Idc is equivalent to g E ZZ, everything is proved.

0

The differential of the adjoint representation of G (in the sense of Definition 4) is a representation of the Lie algebra g which can now be expressed by the commutator.

Theorem 9. The differential ad := Ad. : g -. gl(g) of the adjoint representation is a homomorphism of Lie algebras determined by the formula

ad(X)(Y) = [X, YJ. Proof. By the definition of the differential, we have

Ad(exptX) = exp(tAd.(X)) = 1 + tAd.(X) + ... ; hence

Ad.(X)(Y) =

Xd

Ad(exptX)(Y) - Y

li.o The flow corresponding to the vector field -X is (bt = R P(_tX), since d4ie(e)

dexp(-tX) I

dt LO t=o Applied to a left-invariant vector field Y, however, its differential coincides with Ad(exptX)(Y),

Ad(exptX)(Y) = dL P(tx)dR P(-ex)(Y) = dR p(_tx)(Y) = d4it(Y). Thus the original identity can be rewritten as

Ad.(X)(Y) = li o

`ht(Y) - Y

The right-hand side is precisely the definition of the commutator [Y, -X] _ O

[X, Y].

This representation will also be called the adjoint representation (this time of the Lie algebra g). In case of doubt, the context has to decide whether the representation of the Lie group or that of its Lie algebra is meant. Remark. The definition of the differential immediately implies the identity

Ad(expX) = exp(adX).

6.3. The Adjoint Representation

225

This has to be understood as an identity of operators. Applied to an element Y, it means ad(X)3i3(Y)

exp(X) Y exp(-X) = 1 + ad(X)(Y) +

+ ...

= 1 + [X, Y1 + [X, [2I Y11 + [X, [X 3'XI Y111 + ... .

Example 14. Let g be the three-dimensional Lie algebra which is abstractly defined by the following commutator relations for a basis el, e2, e3: [el, e21 = e3,

[e3, ei1 = e2.

[e2, e31 = el,

The representing matrices of the adjoint representation with respect to this basis can be computed from them. For the operator ad(ei ), we obtain 0

ad(ei)

e2 e3

e3

i

l

=

rO

-e2

0

1

-1

0

e2 e3

Ll

e2 e3

and similarly for the other two operators,

0-1 L2 := ad(e2)

0 rol

0

0 0

10

0 L3 := ad(e3) _

,

1

0

0 0

00

Let us look, on the other hand, more closely at the orthogonal group O(n, R).

It was defined as the set of matrices satisfying f (A) = AAt - 1, = 0. The differential of this map at the point X is

df(A)x = AXt+XAt, and hence, according to Theorem 5 in §3.2, the Lie algebra of O(n, R) is

o(n, R) = Te O(n, R) = {A E M,(R) : A + At = 01. The Lie algebra of the orthogonal group consists precisely of the skewsymmetric matrices, which for n = 3 is apparently spanned by Li, L2 and L3. This proves that the three-dimensional defining representation of o(3, R) is isomorphic to the adjoint representation. In higher dimensions, this fact no longer holds as a simple dimensional consideration shows: A skew-symmetric matrix has exactly as many degrees of freedom as entries above the diagonal. Therefore, 22

dim o(n, R) = and this is equal to n only for n = 3.

- n = n(n2 1)

6. Lie Groups and Homogeneous Spaces

226

Exercises SL(2, R) is not surjective. Hint: 1. The exponential map exp : si(2, R) What are the values that tr exp(A) can attain for A E al(2, R)?

2. Hamilton's quaternions H (Example 9) form not only a vector space, but also a (non-abelian) division algebra, i. e., an associative algebra in which each non-trivial element is invertible. Prove that the standard basis E, I, J, K of Hamilton's quaternions obeys the following algebra relations:

I.K=-J, and compute the inverse of the quaternion h =

0.

3. Identify the quaternions of trace 0,

Ho = {xi . I + x2 J + x3 K j xi, x2, x3 E R}, with the 3-dimensional euclidean space R3. Prove that, for every U E SU(2), the map UxU-1, Ho - Ho, x

defines a special orthogonal transformation of R3. The resulting map e SU(2) - SO(3,R) is a representation which is not injective ("faithful"). 4. The defining representation of SL(2, R) on R2 is the usual matrix action on vectors,

gV

[cx + dy] Ic d] [y] Let V,, be the (n + 1)-dimensional vector space of homogeneous polynomials of degree n in the variables x and y. Define an action of g E SL(2, R) on the polynomial p E V,, by

P(g) ' P ([X])

= P\g_1

[;]).

Prove that B is a representation of SL(2, R).

5. Let G be a Lie group, and let H be a discrete, normal subgroup of G. Prove that H is necessarily contained in the center of G.

Exercises

227

6. Let (p, V) be a representation of the group G. A subspace W of the representation space V is called invariant if. for every g E G. the relation p(g)W C W holds. For trivial reasons, the subspaces W = V and W = {0} are invariant: if the representation has no further invariant subspaces, it is called irreducible. Consider the following two-dimensional representation of the additive group R:

tl

rrI LOW = L0

1

11

Prove that this representation is not irreducible. Does the invariant subspace have an invariant complement?

7. By Theorem 6. the differential of a representation (p. V) of the Lie group G is a representation (Lo., V) of its Lie algebra g. The tensor product of two representations (p, V) and (µ, W) of G is defined by (p ®Fr)(9)(v $ w) := p(9)v (9 p(g)w, g E G. V E V. W E W . Prove that this determines a representation of G on V O W with differential

(e® i),(X)(v ®w) := p,(X )v ®w + v ®p.(X)w, X E g. 8. In order to describe the hyperbolic plane as a homogeneous space. it is useful to introduce a new model for it, the open unit disc.

a) Let D = {z E C I lzj < 1} be the open unit disc with metric

9 = (1_1212)2 I0 Show that the Cayley transform.

I.

`

-i`+i=:x+iy,

c(z) =

c:

i

z-i

is an isometry between D with the metric above and the upper half-plane ?{2 with the fourfold of the hyperbolic metric. b) Let the Lie group

SU(1.1) :_ { act on D via the formula a

b

lb

b

z

:

1a12

- 1b12 = 1}

_ az+b az+b

a Prove that this action is transitive, and that the isotropy group of zero, b

Go := {g E SU(1,1) : g 0 = 0}, is isomorphic to SO(2,IR). Hence D 5 SU(1,1)/SO(2,1R).

Chapter 7

Symplectic Geometry and Mechanics

7.1. Symplectic Manifolds Riemannian geometry is the geometry of a symmetric, bilinear form depending on the point of a manifold. The curvature is a measure of how far two symmetric bilinear forms differ locally. Contrary to this, symplectic geom-

etry is that of an antisymmetric bilinear form depending on the point of a manifold-hence the geometry of a 2-form w. It turns out that all symplectic manifolds are locally equivalent: there cannot be any concept similar to curvature in the sense of Riemannian geometry. Symplectic structures differ, if at all, only globally. Historically, the formulation of mechanics in the sense of Hamilton led to symplectic geometry, hence its essential role in modern mathematical physics.

Definition 1. A symplectic manifold is a pair (1112., w) consisting of a manifold 1112ni of even dimension together with a closed non-degenerate 2form w,

d w = 0 and

A

called the symplectic form or symplectic structure. By Theorem 16 in §3.4, every symplectic manifold is orientable. The volume form is understood to be the 2m-form dM2m

= (-1)

-(--1)/2 '

win

.

m! 229

7. Symplectic Geometry and Mechanics

230

Example 1. In 1R2"' with coordinates {q1, ..., q,", pi, ..., p,,,), the formula m

E dpi A dqi i=1

defines a symplectic form with highest power

w"' = m! -

(-1)-(--1)/2

- dpl A ... A dpm A dq1 A ... A dq,,, .

The volume form in the sense of symplectic geometry is the ordinary volume form of ]R2i'. This symplectic structure is called the canonical symplectic structure.

Example 2. Define a 1-form 0-the so-called Liouville form--in the cotangent bundle T'X"' of an arbitrary m-dimensional manifold as follows: Let V E T,, (T' X ') be a tangent vector at q E T' X' and represent it by a curve

V : (-e, e) - T* X' such that

V(0) = q,

V(0) = V.

Project this curve first by means of the projection 1f : T'X"' - X"' to the manifold, and, after that, apply the 1-form q to the tangent vector of the projected curve:

9(V) := n

d d (T ° V (t)) I=o

The 2-form w := dO is a symplectic structure on T'X'. Any system, {q1, ... , qm }, of coordinates in X' determines-representing a 1-form q as q = >2 pi dqi-coordinates {qt, ... , 9m, pi, ... , p,n } in TX". By the definition of the Liouville form 0, we have m

0= and the 2-form

pi dqi

m

w=d9=EdpiAdgi i=1

is non-degenerate. In particular, the (co-)tangent bundle of every manifold is an orientable manifold (see Exercise 9 in Chapter 3). Further examples of symplectic manifolds arise as the orbits of the coadjoint

representation of a Lie group G. Starting from the adjoint representation, GL(g), of the group G and passing to the dual of the linear Ad : G operator, Ad'(g) := (Ad(g-1))* : g' - g', we obtain a representation

Ad` : G -b GL(g')

231

7.1. Symplectic Manifolds

of the group G in the dual space g' of the vector space g. Through each functional F E g' passes an orbit 01(F) := {Ad*(g)F : g E G}, on which the group G acts transitively. The isotropy group

GF := {g E G : Ad'(g)F = F) is a closed subgroup of G, and 0 *(F) is diffeomorphic to the homogeneous space G/GF. Its Lie algebra can be characterized by a similar condition:

Theorem 1. The Lie algebra OF C g of the isotropy group GF C G is equal to

OF = {XEg :F([X,Y])=0 for all YEg}. Proof. Suppose that F([X,YJ) = 0 holds for all elements Y E g. Then from

(Ad*(exp(t X))F)(Y) = F(Ad(exp(- t X))Y) = F(exp(- t ad(X))Y) =

F(Y - t

z

we immediately obtain Ad*(exp(t X))F = F. The one-parameter group exp(t X) is a subgroup of GF, and hence its tangent X belongs to the Lie algebra OF. This proves one inclusion, the converse is proved analogously.

0 If a Lie group G acts smoothly on a manifold Mm, we can associate with each element X E g of the Lie algebra the unique vector field k on Mm whose integral curves coincide with the trajectories of the one-parameter transformation group exp(t X):

X(x) _

d

The vector field X is called the fundamental vector field corresponding to the element X E g of the Lie algebra. If G acts transitively on Mm, every tangent vector V E TXM'" at a fixed point x E Mm can be realized by a fundamental vector field. This general construction will now be applied to an orbit O' of the coadjoint representation. First, realize a given vector V E TFG' as the value of a

fundamental vector field, f(F) = V. For a further element Y E g of the Lie algebra such that k(F) = V, the equality (X-- Y)(F) = 0 immediately implies

0. e=0

7. Symplectic Geometry and Mechanics

232

Theorem 1 implies X - Y E OF, and the resulting map is injective,

9/9F3X- X(F)ETFO'. For dimensional reasons, it is bijective: the tangent space TFO* to an orbit 0* at F E O' can be identified with the vector space 9/9F. We now define

a symplectic structure wo on each orbit O' C g'.

Definition 2. Let V, W E TFO' be two tangent vectors to the orbit at F E O', and choose elements X, Y E g with f ((F) = V, Y(F) = W. The value of the Kirillov form wo on the vectors V, W is determined by the formula

wo (V, W) := F([X, Y])

Theorem 2. The pair (O',

is a symplectic manifold, and the 2 form

wo. is G-invariant.

Proof. Note first that the 2-form wo is uniquely defined. If the elements X, X1 E g realize the vector V at F, then the difference X - X1 belongs to the Lie algebra OF, and Theorem 1 implies

F([X,Y1) = F([X -X1,Y])+F([X1,Y]) = F([X1,Y]). Moreover, wo is a non-degenerate 2-form. If, in fact, wo (V, W) = 0 for

every tangent vector W E TFO', then we obtain F([X,Y]) = 0 for all elements Y E g. By Theorem 1, X lies in the Lie algebra OF, and hence V = f(F) = 0. It remains to show that wo is a closed form. For two elements X, Y E 9, the function wo (X, k) : O' R is determined by the formula

F([X,YI) Differentiate this relation in the direction of a third fundamental vector field:

2(wo.(X,Y))(F) = d [Ad'(exp(-t.Z)F)[X,Y]]It_o = dtF(Ad(exp(tZ))([X,Y]))It=o = F([Z, [X,YII) Then the expression for the exterior derivative dwo of the 2-form vanishes identically:

dwo. (X, Y, Z) = X (wo (Y, Z)) - Y(wo (X, Z)) + Z(wo (X, f))

-wo.([Y,Z1,X), since it reduces to the Jacobi identity of the Lie algebra g.

0

Corollary 1. Each orbit O' C g' of the coadjoint representation of a Lie group is a manifold of even dimension.

7.1. Sylnplectic Manifolds

233

Example 3. The affine group of R has the matrix representation

G=

{ [0

1 1

a>0,bER}

with Lie algebra 9

=

1[Y l

0

0

:x ,y ER}

.

J11

The computation r[0 0]'[100a -b/a 1 _ 10 ay0bx1 Ad [0 1] [0 1] [0 0] implies that g has one-dimensional orbits. To determine the orbits of the

coadjoint representation, write any element of g' as a pair (a, 0) of real numbers, whose evaluation on the element (x, y) E g is ((a, /3), (x, y)) = ax + /3y

By definition, the group element g =

10

.

1] acts as follows:

(Ad* (g-1)(a, 3), (x, y)) = ((a, R), Ad(g)(x, y)) = ((a, Q), (x, ay - bx)) = ax + /3(ay - bx) = ((a - fib, /3a), (x, y)) . Summarizing, we have Ad*(g-1)(a,0) = (a - f3b,,Oa), and hence for 3 96 0 the coadjoint orbit through (a,,3) is two-dimensional. This example shows that the adjoint and the coadjoint representation of a group G are, in general, not equivalent.

After having discussed examples of symplectic manifolds, we now want to introduce the symplectic gradient, which is the analogue of the gradient of a function on a Riemannian manifold. In this situation, we will make use of the fact that the non-degenerate 2-form w also provides a linear bijection between the tangent bundle TA12m and the cotangent bundle T* M2",.

Definition 3. Let H : M2,

IR be a smooth function on a symplectic manifold. The symplectic gradient s-grad(H) is the vector field on M2in defined by

w(V,s-grad(H)) := dH(V). Example 4. Let {q1, ..., q,,,, p1i ... , pm } be coordinates on M2'" such that the symplectic form w can be written as w = E dpi A dq;. Then

s-grad(H) =

(aH 0

aH

a

This formula immediately follows from the equation defining the symplectic gradient. A curve -y(t) in the symplectic manifold M2rn, represented in the

7. Symplectic Geometry and Mechanics

234

fixed coordinates y(t) = {q1(t), .... q,n(t), pl (t), ... ,p,n(t)}, is thus an integral curve of the vector field s-grad(H) if and only if the so-called Hamilton equations hold:

aH

9i = api

aH and Pi=-aqi

Theorem 3 (Liouville's Theorem). Let H be a function on a symplectic manifold (M2m, w), and suppose that s-grad(H) is a complete vector field with flow 4Dt : M2m , M2m. Then: (1) The Lie derivative of w vanishes,

Gs-g,ad(H)(w) = 0. (2) The flow 't preserves the symplectic volume,

J

dM2m =

dm2m.

J

Proof. From dw = 0 and Theorem 32 in §3.9, we conclude that

Gggrad(H)(w) = d(s-grad(H) J w) = - dd H = 0. ,11m

do not alter the symplectic structure, i.e., 4 (w) = w, and so both assertions are proved. 0 Hence the diffeomorphisms 4)t : A f2m

The existence of this invariant measure has consequences for the dynamics of symplectic gradient fields.

Theorem 4 (Poincare's Return Theorem). Let (M2,, w) be a symplectic manifold of finite volume, and let 4bt : M2m _ hf2m be the flow of the symplectic gradient of a smooth function H. For any set A C M2m of positive measure, the set

B = {x E A : tn(x) 0 A for all n = 1, 2, ...} has measure zero.

Proof. Note first that the intersections 4>_n(B) fl B are empty. Any point

xE

fl B would be a point x E B such that 4n(x) E B C A,

contradicting the definition of the set B. This immediately also implies that

the intersections ' _n(B) fl 4'_m(B) are empty for n

m, and, from the

invariance of the measure, we obtain

x

oc

Evol(B) = Evol((Dn(B)) < Vol(M2m) < x, n=1

n=1

i. e., the measure of the set B has to vanish.

0

This result has several famous generalizations, as the only example of which we quote Birkhoff's ergodicity theorem (without proof).

?.1. Symplectic Manifolds

235

Theorem 5 (Birkhoff's Ergodicity Theorem). Let f be an integrable function on the symplectic manifold (M2,, w), and let Ot be the flow of a symplectic gradient. Then the following limit exists almost everywhere: lim 1 t

Jo

t

f o .0,(x) =: f` (x) .

Furthermore, the function f' is also integrable, and its integral coincides with that of f. Lastly, f* is invariant under the flow fit. Definition 4. The Poisson bracket of two functions f and g on a symplectic manifold is the function

{f,9} = w(s-grad(f),s-grad(9)) = dg(s-grad(f)) = -df(s-grad(g)). Example 5. In the {q, p}-coordinates, we have

{f, 9} _

m Of a9

_ Of a9

5q1

aq1 Opi

i=1 \ apt

In the next theorem, we summarize the properties of the Poisson bracket.

Theorem 6. The ring C'°(M2,) endowed with the Poisson bracket is a Lie-Poisson algebra: for constants cl, c2 E R;

(1)

(2) { f, g} = -{g, f }; (3) { f, {g, h}} + {g, {h, f }} + {h, { f,g}} = 0

(Jacobi identity);

(4) {f,g-h} =g {f,h}+h- {f - g}; (5) s-grad({f, 9}) = [s-grad(f ), s-grrd(9)J.

Proof. The two first identities result immediately from the definition of the Poisson bracket, and the fourth follows from d(g - h) = g - dh + h dg. We prove (5). We insert the vector fields V := s-grad(f ), W := s-grad(g) together with an additional vector field Y into the equation defining the 3-form dw,

0 = dw(V,W,Y) = V(w(W, Y)) - W(w(V, Y)) + Y(w(V, W)) - w([V, W], Y) + w([V, YJ, W) - w([W, Y), V) .

Applying the definition of the symplectic gradient as well as that of the Poisson bracket, we can rewrite this equation as

0 = -V (Y(9)) + W (Y (f )) + Y({ f, 9}) - w([V, W], Y)

+ [V,Y)(9)-[W,Yj(f) = -Y({f,9})+w(Y,[V,WI)

7. Symplectic Geometry and Mechanics

236

Hence s-grad({ f, g}) = [V, W] = [s-grad(f ), s-grad(g)J. The Jacobi identity is a consequence of formula (5). In fact, we obtain

{f. {g,h}}+{g.{h, f}}+ {h.{f,g}} = s-grad(f)(s-grad(g)(h)) - s-grad(g)(s-grad(f)(h)) -s-grad({f,g})(h) = [s-grad(f ), s-grad(g)] (h) - [s-grad(f ), s-grad(g)] (h) = 0. A Hamiltonian system consists of a symplectic manifold (111211, W, H) to-

gether with a function H. The integration of the corresponding Hamilton equation relies on determining the integral curves of the vector field s-grad(H). For this, there exists an analogous notion of first integrals as in the Riemannian case.

Definition 5. A function f : 1112" -. R is called a first integral of the Hamilton function if it is constant on each integral curve of the vector field s-grad(H).

Theorem 7. (1) A function f is a first integral of the Hamilton function H if and only if its Poisson bracket with H vanishes,

{ f, H} = 0. (2) The set of all first integrals of a Hamilton function is a Lie-Poisson algebra.

Proof. We compute the derivative of a function f along an integral curve y(t) of s-grad(H):

dt f o -y(t) = df (j(t)) = df (s-grad(H)) = {H, f } . This implies the first assertion. The second follows from the Jacobi identity for the Poisson bracket.

7.2. The Darboux Theorem The Darboux theorem states that all symplectic manifolds are locally equivalent.

Theorem 8 (Darboux Theorem). Near each point x E 1112n of a symplectic manifold (M2i/.w), there exists a chart h : U C 1112m - R2m in which the symplectic form w is the pullback of the usual symplectic form.

w = h*

dpi n dqi) e=1

7.2. The Darboux Theorem

237

Coordinates with these properties are called symplectic (canonical) coordinates.

Proof. In the cotangent space TTM2," to the manifold at the point x E M2",, we choose a basis al, ... , o , ILl...... m in which the symplectic form at this point is represented in normal form, Iii

w(x) = E ai A pi . i=1

Consider, moreover, a chart (D : V - 1R 2»i around x such that

4t(x) = 0 and w(x) = 4D` (dPAdQi(0)) Denoting the corresponding symplectic form on V by

Wi := fi' I

dpi A dqi I

,

i=1

we see that there exists a neighborhood U C V of x such that for all parameters t E [0,1] the form

wt :_ (1 - t)w + twl is a symplectic structure on U. In fact, dwt = 0, and since wt(x) = w(x) for all t, a compactness argument shows that all the forms wt (t E [0,11) do not degenerate at the same time in a neighborhood of x. Since d(wl - wo) = 0, Poincare's lemma shows the existence of a 1-forma such that the difference

w, -wo = da is the exterior derivative of this 1-form. By subtracting locally, if necessary, a 1-form with constant coefficients from a, we may assume that a vanishes at the point x, a(x) = 0. Dualizing a by means of the symplectic forms wt, we obtain a family Wt of vector fields on U parametrized by t,

wt W, Wt) = a(V) .

Let cp(y, t) E Mgr" be the solution of the (non-autonomous) differential equation Ve(t) = Wt('p(t)), V(0) = y All the vector fields W1(x) 0 vanish at the point x, and the solution

corresponding to the initial condition x is constant, p(x, t) - x. Hence there exists a neighborhood U1 C U of x such that, for every initial condition y E U1, the corresponding solution p(y, t) is at least defined in the interval

7. Symplectic Geometry and Mechanics

238

[0, 1]. Let t : Ul -+ A12°' be the corresponding map. The formula for the Lie derivative of a differential form (Theorem 32, §3.9) implies

(L)

+ 0i ('Ca , ,a (Wt)) = V P1 - w) + Vi (Gww, ((WO) dt 4 (wt) _ 'pi = ipi (wi - w + d(Wt J wt) + W1 J dwt) = tipi (wt - w - da) = 0.

Thus Spi (wl) = pp(w) = w, and 4 o cpl is the chart we were looking for,

(DoVI)" Edpindyi

w.

O

7.3. First Integrals and the Moment Map As in the Riemannian case, some first integrals can be derived from symmetry considerations. The isometries (which are not available on symplectic manifolds) giving rise to these first integrals will be replaced by symplectic diffeomorphisms which, however, satisfy a compatibilty condition with respect to the Hamilton function under consideration. We will describe this in detail in the case of an exact symplectic manifold, i. e. , a symplectic manifold whose symplectic form is the exterior derivative of a 1-form, w = d9.

Suppose that a Lie group G acts from the left on M2" in such a way that each diffeomorphism 1g : M2i' -+ M2ni leaves the form 9 invariant,

l9(0) = 9. These diffeomorphisms 1g are then symplectic, i.e., they preserve the sym-

plectic structure w. Now let X E g be an element of the Lie algebra of the group G, and let k be the fundamental vector field corresponding to X under the G-action on M2in. The evaluation of the 1-form 9 on X is a function,

.6(X) := 0(.k). This construction determines a linear Map '1 : g _ Coo(M2,n) from the Lie algebra g to the space of functions COD(M2in) on the symplectic manifold. Its properties are the subject of the following symplectic variant of Noether's theorem.

Theorem 9 (Noether's Theorem). (1) $ : g - C, (M2,) is a homomorphism of Lie algebras, 'DQX, Y]) = (4'(X ), CY)} . (2) s-grado4' corresponds to the transition to fundamental vector fields,

s-grad(qD(X)) = f(.

239

7.3. First Integrals and the Moment Map

(3) If the Hamilton function H is G-invariant, then

is a first

integral of H, 0.

Proof. Fix an element X E g in the Lie algebra and consider the oneparameter group of diffeomorphisms corresponding to the group elements exp(-t X). Its generating vector field is the fundamental vector field X. The relation l9 (0) = 0 implies that the Lie derivative of 0 with respect to X vanishes,

0 = cX(0) = X-jd0+d(XJ0) = X_jw+d(-t(X)). Thus, for every vector field V, we have the equation

-w(X,V) = V(4(X)) = w(V,s-grad(4'(X))) as well as X = s-grad(4(X)). Using this formula, we compute the difference

{4(X),4(Y)} - .0([X, Y]) = w(X,Y) - 4,([X, Y]) = dO(X,Y) - 4,([X, Y])

= X(4(Y)) -Y(,t(X)) -24([X,Y]) = 2({'F(X), 4'(Y)} - -NX,Y])) , and this yields

{4(X), F(Y)} = C[X, Y]) . If, finally, the Hamilton function is G-invariant, we obtain

-X(H) = 0,

{H,fi(X)} =

0

i.e., 4(X) is a first integral.

The elements of the Lie algebra g provide first integrals for every G-invariant Hamilton function. These first integrals can be combined into a single vector-valued first integral by passing to the dual space g'.

Definition 6. The moment map of a Hamiltonian system with symmetry group G is the map IF : M2m -+ g' from the symplectic manifold to the dual of the Lie algebra defined by

CX)(m)

IF(m)(X)

Theorem 10. (1) The map %P is Ad'-equivariant, i. e., the following diagram commutes: M2m

19

M2.

7. Symplectic Geometry and Mechanics

240

(2) %V is a first integral of H.

Proof. The fundamental vector field X of a G-action has the following invariance property: d X (1,(x)) = dt [exp(-tX) g x] Jt_o _

d lg

d

dt

[exp(-t Ad(g-1)X)

xJ I t-0

= dlg(Ad(g-1)X(x)).

N

N

The 1-form 0 is G-invariant by assumption, and from this we obtain

'P(lgx)X = 8(X(lg.x)) = B(Ad(g-1)X(x)) = '(x)(Ad(g-1)X). 0 In Exercise 11, we discuss the case M'' = T'R3 with symmetry group SO(3, R) and its usual representation on R3. In particular, it is shown that the moment map P : T`R3 --+ so(3, R) = R3 coincides with classical angular momentum, hence justifying its name. Closely related to this situation is the following example:

Example 6. Consider the 2-dimensional representation of G = SL(2, R) on

M2 = V = R2. Its cotangent bundle is T'M = V x V' ^' V x V, since the representation V is self-dual. A group element g E SL(2, R) acts on an element (p, q) E V x V of the cotangent bundle by g - (p, q) = (gR gq)We call two elements (p, q) and (p', q') equivalent if they lie in the same G-orbit, (p, q) - (p', q'). Let p = (pl, p2) and q = (ql, q2) be the components of the vectors p and q, respectively. One easily computes that the moment map is given by V x V - + s1(2, R),

2 (g1P2

(q, p) --

+ g2P1)

q2P2

[

1

-gipi

- 2(g1p2 + g2Pl)

In particular, this map is equivariant with respect to the adjoint action of SL(2, R) on sl(2, R). The moment map is best studied by examining its action on SL(2, R)-orbits. For this, observe that the quantity

det(q, p) := det [qi pi g2

= g1P2 - 92P1

P2

is SL(2, R)-invariant. It thus allows to parametrize the orbit space; we omit the easy proof here'.

1For details on this SL(2,R)-action, we refer to §1.4. of the book by Hanspeter Kraft, Ceometnsche Afethoden in der Invariantentheorie. Vieweg, 1985.

7.4. Completely Integrable Hamiltonian Systems

241

Theorem 11. (1) If det(q,p) _: A 56 0, then

(q, p) -

0 A ( (1)1(0))

(2) det(q,p) = 0 if and only if p and q are linearly dependent. In this case, there exist infinitely many G-orbits, for which one can choose the following representatives: 0 0 0 0 0 0 ( (0)1(0)), ((,U) W) ((1)1(0)),

with AERR.

Thus, the moment map acts on G-orbits as follows:

((01),(A0))

LA02

-A/2J'

\(0)' 0// ~

[0

0

J

((0) ' (1)) ' (CO) ' Co)) [0 0 In particular, the generic orbits with parameter A 54 0 are mapped to semisimple elements of the Lie algebra sl(2, R). Their orbits are 2-dimensional closed submanifolds of sl(2, J). 7.4. Completely Integrable Hamiltonian Systems In this section we will make use of the following fact concerning the structure of discrete subgroups IF of the additive group IRA.

Theorem 12. Let t C ]RI be a discrete subgroup. Then there exist linearly independent vectors vl, ... , vk such that

r=

k

m; vi

:

m; an integer

i. e., F is the lattice generated by the vectors vi, .... vk.

Proof. If I' # {0} is not trivial, we choose a vector ryl E r such that I I7i I I 0 0 and consider the ball D" (0; I I7i I I) The intersection D" (0; I I71 I I) nr

is a compact and discrete subset of IR^, hence finite. Thus, on the straight line generated by 71i there exists a vector 7i E D° (0; 117, 1I) n r realizing the minimum of the distance to 0 E lR'. For this vector we have Ht

7i n r = {m 7j : m an integer),

since any vector x 36 m7i belonging to the intersection (1R 7i) n r would have to lie in one of the segments and then (m+1)7i -x would be a vector on the line through 7t with a smaller distance to 0 than 7l*. If the group r contains only integer multiples of -y,*, the proof is completed.

7. Syrnplectic Geometry and Mechanics

242

Otherwise, there has to exist a vector 1'2 E r\{m - ryi : man integer}. We project ry2 orthogonally to the straight line passing through ry1 and denote by Y2 the resulting vector. It lies in one of the half-closed segments y2 E [m - 'y , (m + 1) ryi ). Let E be the cylinder with axis [m' . (m + 1) - 7i) whose radius is equal to the distance from ry2 to the line through 'Y1. In this cylinder, there are again only finitely many elements of the group r. Let rye be the vector in r n E whose distance to the axis of the cylinder is minimal and which is not a multiple of ryi . Then we have 2

r n {R 7i ED R7;)

mi ry,

:

mi an integer

.

i=1

In fact, if there were a point x 0 miel + m2e2 in r n {Rryi ® R-y }, then x would belong to the interior of a parallelogram in the {'y , 7s }-plane. Taking the difference with a vertex of this parallelogram, we obtain a vector in r lying closer than -y; to the axis of the cylinder. Repeating this construction finitely many times proves the assertion. 0

Corollary 2. Let F C R" be a discrete subgroup. Then R"/r is diffeomorphic to the product of a k-dimensional torus Tk with R"-k,

R"/r

Tk x

Rn-k

Theorem 13 (Arnold-Liouville Theorem). Let (M2m, w, H) be a Hamiltonian system, and let fl = H, f2, ..., fm be m functions with the following properties:

(1) all functions fi are first integrals of H: (2) the functions fi commute, { fi, f;} = 0; (3) the differentials dfl, ..., df,,, are linearly independent at each point; (4) the symplectic gradients s-grad(fi) are complete vector fields.

For a given point c = (cl, ... , c,,) E R'", we consider the level manifold

Al, = {x E M2m : fl(x) = C1, ..., fm(x) = C,n}. Then:

a) The connected components of Al, are diffeomorphic to Tk x

R'-k

b) The vector field s-grad(H) is tangent to Mc. In particular, each integral curve of this vector field is completely contained in one of the level manifolds. c) If Mc is compact and connected, angle coordinates y91, ... , cp,,, can be introduced in Mc -- T' so that the integral curves of s-grad(H)

7.4. Completely Integrable Hamiltonian Systems

243

are described by the system of differential equations

y , = vi,

v; = constant.

Proof. Consider the flows 4i , ... , 4if

:

M2m

M2, of the symplectic

gradients s-grad(fi). Since

0 = s-grad{fi,f3) = [s-grad(fi),s-grad(f3)1, all these flows commute with one another (Theorem 36, §3.9). This determines an action of the additive group R' on the manifold M2'":

(tl, ... , tm) x = .01 0 ... The orbits of this R'-action coincide with the connected components of the level manifolds. In fact, since

0 = {ff,fi} = s-grad(fi)(f3), each function fi is constant on every orbit. Hence, the orbits of the R'"action are contained in the level manifolds. On the other hand, both are m-dimensional submanifolds of M2in, since the differentials dfl, ..., dfm are linearly independent. The isotropy group r(xo) = {t E Rm : t xo = xo} of a point xo E M2rn for the R'-action is discrete. Hence each component of a level manifold is diffeomorphic to the product of a torus and euclidean space:

Rm/r(xo) = T" x ][ Assertions a) and b) are proved, so we turn to the remaining one. Suppose that a level manifold Mc is compact and connected. Choose a point xo E Mf and denote by v1, ... , v,1 E Rm a basis of the isotropy group r(xo). Representing the basis vectors {v;) of the vector space I(tm as linear combinations of the vectors in the standard basis e1, ... , em of R.. M

vi = 1: ajaeo , a=1

we obtain a quadratic matrix A := (aid). Let m

r(m) _

{ni.ei

:

ni an integer

i=1

denote the standard integral lattice in Rm. Then m

0: Rm/r(m) -. Rm/r(xo) _ -

m

o Exi. ei) = E xi . Vi, i=1

defines a diffeomorphism. The inverse of this map, 4D-1 : MM -+ Rm/r(m) = S1 x ... x S',

i=1

7. Symplectic Geometry and Mechanics

244

as well as its components, 4b-1 = (WI, ..., cp"), lead to the angle coordinates

for the level manifold M, In fact, if yl, ... , y' are the coordinates in R"'/r(xo) = MM determined by 1

m

M. VM,

1

then, by the construction of the R'-action on Af,

s-grad(fi) = y; With respect to the

gyom}-coordinates, this yields m a aj,

s-grad(fi) = a=1

awa

In particular. the symplectic gradient s-grad(H) is a vector field with constant coefficients on the torus MM = T', and the third assertion results by taking v. := alb. W e want to discuss more closely h o w the angle coordinates ( 1, ... , m) of a compact and connected level manifold can be determined directly from the commuting first integrals. This will lead to an explicit algorithm for the integration of a Hamiltonian system (M2m, w, H) provided with sufficiently many commuting first integrals. Because of this procedure, these systems are called completely integrable (or integrable by quadrature). Denote by wl (c), ... , ul,,, (c) the frame of 1-forms on Al, dual to the vector fields s-grad(fl ), ..., s-grad(f,,,). The representation of the vector fields s-grad( f;) in terms of the vector fields 8/&pj immediately implies the following formula for the differentials: M

d'pi =

=1

aia wa(c)

Let ik be the closed curve in MM corresponding to the parameter values rpm=0. Then m

aik = f dcpi = F, ai0 f wn(c) 'Y k

a=1

k

Hence, first the coefficients aid and then the angle coordinates can be computed directly from the first integrals. We summarize this in the form of an algorithm comprising five steps.

Step 1. Fix c = (cl, ... , c,,,) E IIt"' and let Al, be compact and connected. Choose a homology basis y', ... , y", for the first homology group H, (Al,: 7L).

7.4. Completely Integrable Hamiltonian Systems

245

Step 2. Compute the symplectic gradients s-grad(fi) of the first integrals

f1=H,f2.....fm Step 3. Determine the frame of 1-forms w1 (c), ..., wm(c) dual to the frame of vector fields s-grad(f l ), ... , s-grad (f.. ) on M.

Step 4. Compute the line integrals frk w0(c), and invert the resulting (m x m) matrix. This yields the matrix A = (aij(c)). Step 5. Compute the angle coordinates (pi(c) on the level set MM from the equations m

dvi = E ai0(c) wa(c), 1 < i < m. Q=1

This procedure computes the angle coordinates Vi(c) on one level manifold Mc. Note that, according to Step 5, these are only determined up to constants. As we vary the parameters c = (cl, ... , c,,,), the Cpl, ..., cp,,, become functions on an open neighborhood of a level manifold Mc C M2m. Since the symplectic gradients are tangent to Mc, the Poisson bracket with the original functions f1, . . . , fm is computed by

IVi,fjI = dpi(s-gradfj) = aij(fl, . . . , fm). Moreover, it is a function exclusively depending on fl, . . . , fm. Similarly, we prove that the functions {Vi, cpj } are also constant on the level sets.

Lemma 1. The Poisson brackets {cpj,Vj} = bij(fl, ...,fm) are functions depending only on fl, ... , fm. In particular, they are constant on each level set 't1c.

Proof. We compute the derivative of {vi, wj } with respect to the vector field s-grad(fk):

{{Vi,'j},fk} _ {Wj,fk},'Pi} - {{fk,'ci},'pj} _ -{aik(fl, ...,fm),'Pi} + {aik /lfl, ...,fm),pj} m 49

m COajk a=1 m

Oya

8ajk

fta

}

a_I

as -

aaEk

aj.)

Thus all the derivatives cpj} (1 < k < m) are constant on every torus T' = Afc. But then the Poisson bracket {Vi, oj} itself is constant on every level manifold Mc. 0

7. Symplectic Geometry and Mechanics

246

Now we alter the angle coordinates, which up to now were considered only on a single level manifold, by a suitable constant on adjacent level manifolds. The aim of choosing these constants of integration for the angle coordinates is to obtain functions cpi commuting on M2n'.

... , B," (yl, ... , y') such

Lemma 2. There exist functions Bl (yl, that the angle coordinates

:= Wi+Bi(fl,...,fm)

Bpi

commute on the symplectic manifold M2m, {(pi , Vj*) = 0.

Proof. Using the notation {cpi, f;} = aij and {Wi,co } = bij, we apply the Jacobi identity to the triples (cpi, ip,, fk) and (ipi, Wj, cpk), and take into account the fact that the Poisson brackets { fi, fl) vanish. Thus we obtain the relations m aaik Oa,j m

E

E

a;a-

Obi;

ab;k

y u 'aka +

ft la

= 0,

aia

1

'

afa +8bki d ajQ J = 0 .

Inserting also the coefficients air of the inverse of the matrix A = (aid), we consider the 2-form m m

Il :_

> bijai°a'p dy° A dyO. i,j=1 a,p=1

The above relations say that fl is a closed 2-form. In fact, the first relation means that the 1-forms m

E a° dy°

of

a=1

are closed, doi = 0. Hence we can choose coordinates z1, ... , z'" such that of = dzi, and 1 becomes M

fl = > bij

dzi Adzj .

i,j=1

Since

Obi; Ozk

_

m

1

Obi,

ay°

Oya

8zk

8bij a= 1

or or aka

the second relation then precisely expresses the vanishing of the exterior derivative dfl = 0. The angle coordinates we set out to find are now taken to have the form m Bpi

aiaBa

Wi + O=1

247

7.4. Completely Integrable Hamiltonian Systems

with functions Bl*, ..., Bm depending only on fl, ... , f,,,. Then m

ai.

aY

a;l_ M

B.

syv

L

aB;

a (aiaaj3 - ajasid)

+ a.8=1

Taking into account the first of the relations above, the condition 0 turns out to be equivalent to

bij =

1:

".

a.3=1

a 33

(aiflaja - ai.aj,3)

This system of differential equations can be reformulated as

aB M 8ii - aya a

r3

m

[: bijaiaajZ, i;=1

and, by Poincare's lemma, it has a solution, since the 2-form f) considered above is closed.

Thus we can determine the constants of integration occurring in the transition from the differentials m

d'pi(c) = E afa(r) wa(c) a=1

to the angle coordinates on the individual level manifolds in such a way that the functions Vi, defined in a neighborhood of a level manifold, commute

on M2'". Now we add so-called action coordinates J1, ..., J,, and thereby bring the symplectic structure near a level manifold into normal form. The Hamilton function H = f, is itself constant on the level manifolds and only depends on the action variables, H = H(J1, ...,J,,,).

Theorem 14 (Action and Angle Coordinates). In a neighborhood of any compact, connected level manifold M,: C At" of a Hamiltonian system determined by m commuting integrals fl, ... , fm, there exist angle coordinates Cpl, ... , v,,, and first integrals J1, .... J,,, such that in

w = dindJi. i=1

In particular, this implies

Vj} = 0 = {Ji, Jj} and {y'i, Jj} = bij.

Proof. First, we determine the angle coordinates near the compact, connected level manifold MM such that

J(pj, ypj} = 0 and

{tpi, fj} = aij(fi,

fm)

248

7. Symplectic Geometry and Mechanics

We look for the functions J1, ..., J,,, using the Ansatz J, = Ai(fl, and compute the Poisson bracket m

8Aj ai. OY.

The condition {,..1j} = 8ij leads to the system of equations OA

ayj

= aij

where a'j is the inverse of the matrix aij. By Poincare's lemma, the integrability condition is

8aij

8aik

8yk = $yjj

On the other hand, we obtain from the Jacobi identity 0 = {`r'k,{iPi,.fj}}+{Vi.{fj, 7k}}+{.f .{Y"'k,Y^'i}}, and, taking into account {j,9i.k} = 0, this immediately yields

8aq

E fta aka

= L 8ukj

8 a aip..

a=1

Q=1

y

A simple computation shows that this relation is equivalent to the integrability condition for the coefficients aij of the inverse matrix. 0

Example 7 (Two-dimensional Hamiltonian System). Consider in R2 with the symplectic structure w = dp A dq a Hamilton function H(q, p) for which the level curves (q, P) E 1R2

H(q. p) = c}

:

are closed. The action variable J = J(H) is a function of H, and, together with the angle coordinate , we have

dpAdq = dV AV = Applying the Hodge operator * of R2, we see that dy: is proportional to the 1-form *dH, * dH .

dV

J'(H) IIdHII2 The integral of d,o over each level curve is an integer, which we take to be equal to -1. This condition is called the classical Bohr-Sommerfeld condition. The equation J'(c)

1

nor IIdHII2

* dH

7.4. Completely Integrable Hamiltonian Systems

249

uniquely determines the action variable J = J(H) in terms of the Hamilton function H. Consider the domain bounded by the level curve A4,,

Q, = {(q, p) E R2 : H(9, p) 5 C1. The vector field W := grad(H)/Ilgrad(H)II2 satisfies W(H) - 1; hence its flow +t maps the set flc onto tl +t. We compute the change of the area of the domain: d

(vol(S2c)) =

r

Ji1c

tli o

t (J$2) -J t k

d(W i dlt2) =

st4

J

W I dR2 =

lo

2)

If

,1 4

=

Js24 Gbb'(dR2)

IIdHII2 *

dH = X(c)

.

Thus the action variable J = J(H) can be interpreted as follows: J(c) is the volume vol(Q,) of the domain bounded by the level curve hf, _ {(q, p) E R2 : H(q,p) = c}.

Example 8 (Spherical Pendulum). Consider spherical coordinates on the sphere S2\{N, S} with the north and south pole deleted, h(4,) = (cos cp cos ti, sin yp cos 0 , sin

g) .

The Riemannian metric of the sphere is then described by the matrix (see Example 14 in §3.2) 9

_

1

L

0

O cost'

Denote the coordinates in the cotangent bundle by (cp, o, pw, p v) and consider the Hamilton function

H = 2v+2 H describes the motion of a pendulum of length one suspended in the center of the sphere. The meaning of the angles tp and t'} can be seen in the picture to follow. For simplicity, the gravitational constant was taken to be one.

250

7. Symplectic Geometry and Mechanics

Z

The variable (p does not explicitly occur in the Hamilton function; hence P:= p,, = Ocos2 iii is a first integral, {H, P} = 0. The Hamiltonian system (T*S2, H) is thus completely integrable. The level manifold

M2(ci,c2) :_ {pw,po) E T* S2 : P = c1, H = C2} is empty for negative values of the parameter c2, and it consists of the south pole remaining at rest in the case c2 = 0. Hence we suppose that the parameter is positive, c2 > 0. The equations describing the manifold A12(cl,c2) are

C1 = pr,,

c2 =

2 + 2 cs2 i + 1 + sin ?P.

The relation cl = 0 implies that cp has to be constant. In this case, the second equation of motion reduces to that of a planar pendulum, so we will henceforth exclude this case. Depending upon the sign of c1, the function is monotone increasing or decreasing, i. e., the pendulum does not change its direction of motion. Rewriting the second equation and setting z = sin 4i, we obtain

p=

c2-1-z- 2(1

C- 2

z2)

U(z)

and see that the function U(z) thus defined cannot be negative. The limiting

case p = 0 corresponds to the pendulum moving in a fixed plane, hence on a meridian. To see where the function U(z) can be strictly positive, we multiply it by the denominator 1- z2 and look for the zeroes of the resulting cubic polynomial,

V(z) = (1 - z2)U(z) =

c2 (c2 - 1 - z)(1 - z2) - 2

.

7.4. Completely Integrable Hamiltonian Systems

251

At the boundaries of the interval, V(±1) = -c/2 < 0 is negative, and at +oe the function V diverges to +oc. Hence one of the three zeroes has to lie above 1, and, since it cannot correspond to any angle 1;i, this zero has no physical relevance. In the generic case, the other zeroes belong to the interval (-1, 1); between them V, and hence also U, is positive. We conclude that V(z) has the qualitative behavior of the graph on the previous page. The mass point can only move between the two meridians corresponding to the zeroes in the interval (-1, 1). The boundary values t ,'4'2 defined this way are actually reached at the end of every up or down swing. Summarizing,

the level manifold can be parametrized by the two parameters 4z = V and

0=

via ( P Cl!

cos

- 2 - 2sini ) .

It consists of 2 two-dimensional tori, where ' only takes values in (>G1, 021. The corresponding coordinate vector fields are a a a a apti; a

a = TT' a = a + o '

ap

,

We express the symplectic gradients of the first integrals in the coordinates of the level manifold: P;, a a d P,2. a s-grad(H)

= cost ;, a + p av - a (2 cost V, + 1 + sin V) aplo Pro

a

a

= cost' ap +Pva +Pv a i a a _ 2 Zi app. + Pv a,. pw

s-grad(P) = a =

ate,

.

"PV

7. Symplectic Geometry and Mechanics

252

The dual forms w1(cl , c2) and w2(cl, c2) are thus 1

wl(Cl,C2) =

de*,

PO

C12

w2(cl,c2) = dw - p o Cos '+G dip

.

We compute the periods with respect to the homology cycle 'rl which is parametrized in M2(c1i C2) by 't()'. The factor 2 is a consequence of the fact that the boundaries, y1,'Y2, of the interval correspond to the minimum and the maximum of the motion, whereas a cycle is meant to be a motion between two extremal points of the same kind:

b1l =

f wl = 2 ti

J

%

c>>

Py

b12 =

f

w2 = -2c1

7i

r 02 diP Jv Pb oos2

Let, similarly, 72 denote the homology cycle determined by 0 < gyp` < 2Tr. Then b21 =

Jwi = 0,

Jw2

b22 =

= 27r .

The matrix occurring in the algorithm for computing the angle coordinates is now easily calculated:

-l [b21

- [a21

b221

a22]

1

-j w2

f7twl

2vfnwl

0

1/27r

The resulting quotient of the basic frequencies is

V1 =all =_1f1-'2= 21r V2

a12

,

C1

7r

112

.//ol

di&

cost tai ' PW

This is nothing but the perihelion precession, i.e. the total variation of the angle V for a complete cycle: O +a

dye = 2

d` d1/b = 2

02

. dpi = 2c1

d

27r L1 .

COS2t' ' PG = .1,1 J 'rI J d it fo , fo, I In general, the motion is quasi-periodic. The trajectory is closed if and 1

only if vl /v2 is rational; otherwise, it is dense on the torus. The transition to action angle coordinates allows us to determine the physically relevant basic frequencies of the system without having to explicitly perform the integration. This accounts for their importance in astronomic perturbation theory.

7.5. Formulations of Mechanics Newton's equations describe the motion of a mechanical system under the impact of a force. The latter is understood as a vector field depending upon position and velocity, and is central for Newton's formulation of mechanics.

7.5. Formulations of Mechanics

253

During the 18th century, this view changed in that Lagrange considered the action integral as the fundamental quantity for the description of dynamics. Newton as well as Lagrange formulated mechanics within the tangent bundle of configuration space. In the 19th century, by transition to the cotangent bundle, Hamilton succeeded in formulating the dynamics of mechanical systems within the framework of symplectic geometry. The aim of this section is to explain these fundamental ideas of mechanics and the related mathematical structures.

Newton (1643-1727)

Newtonian systems

Lagrange (1736-1813)

Hamilton (1805-1865)

Lagrangian systems Hamiltonian systems

i Newtonian systems with potential energy

1

hyper-regular Lagrangian systems

Legendre transformations

Mathematical Contents: Riemannian geometry

Finsler geometry

symplectic geometry

Formulation of Mechanics According to Newton In Newtonian mechanics, the state of a mechanical system is described by finitely many real parameters. This leads to the notion of configuration space. which is a smooth and finite-dimensional manifold Mm. A motion of the mechanical system is a curve 'y : (a, b) - M' in configuration space.

Its tangent-the velocity-is then a curve ' : (a, b) - TM' in the tangent bundle. and this space is called the phase space. According to Newton, the forces acting on the mechanical system are described by vector fields depending only on position and velocity, that is, vector fields X on TM"'. However, not all vector fields are allowed, since a force can act only in space.

7. Symplectic Geometry and Mechanics

254

The force vector field X has to satisfy the condition

drr o X = Id. Here 7r : TM" - Mm denotes the projection of the tangent bundle, and dir : TTM' --+ TM' is its differential. Summarizing, we arrive at the notion of a Newtonian system.

Definition 7. An autonomous Newtonian system is a triple (Mm, g, X ) consisting of a manifold M'", a Riemannian metric g, and a vector field on the space TM'" such that d7r o X = IdTMm .

The function T : TM'" -+ R defined by T(v) := 2 g(v,v)

is called the kinetic energy.

Definition 8. A motion of the Newtonian system (M"', g, X) is understood to be a curve y : (a, b) - M' in configuration space whose curve of tangents ry : (a, b) - TM'" is an integral curve of X,

'1'(t) = X MW This is an invariant formulation of Newton's equation.

Example 9. Consider R with the coordinate x and identify TR = R2 with R2. Here the coordinates are denoted by {x, i}. The vector field 8 1 0 2

X=

wax+m(-k x-P)8i

on TR has the required projection property, and a curve x(t) in R is a motion in this Newtonian system if x(t) is a solution of the oscillator equation

ml(t) = -k2x(t) - pe(t) . Example 10. Let (Mm,g) be a Riemannian manifold. We define a vector field S : TM°1 - TTMm on its tangent bundle-the so-called geodesic spray-as follows: If v E TAM' is a tangent vector, then there exists precisely one geodesic line y,,,,, : (-e, c) -> Mm such that

7'r,.(0) = X, % AO) = v. Consider its tangent curve, %,v : (-e, e) - TM'", and set S(v)

dt (%"(t)) t=o

The relation 7r o ryx,,, (t) = ry v (t) implies dir o S = IdTMm. Hence (M', g, S) is a Newtonian system, and the motions of this system are the geodesic lines

7.5. Formulations of Mechanics

255

of the Riemannian manifold (Mm, g). In the coordinates {x', ii} of the tangent bundle, the geodesic spray is given by the formula

S=

x' i=1

axi

- E r;k x'xA axi i,j,k=1

a straightforward consequence of the system of differential equations describing geodesic lines-see §5.7. Many Newtonian systems have a potential energy. This is a smooth function

V : Mm --+ R defined on configuration space. The gradient grad(V) is a vector field on Mm locally determined by grad(V)

'"

av a

- `i,j=1 g'' (x) axi axj

In the sequel we will need, however, a different vector field, denoted by grad(V). This will be a vector field on phase space. At the point v E TM'", it is defined by the following equation: d

grad

(v + t - grad(V)(ir(v))) e-o

(V) (v) = dt In the manifold Mm, the curve v + t - grad(V)(a(v)) projects to the base point rr(v) E M' of the vector v E TM'. Hence the vector field grad(V) projects to zero under the differential dir, d7r o grad(V) = 0,

and, for any potential energy V, the vector field X := S - grad(V) is an admissible vector field on TM'" in the sense of Newtonian mechanics. In local coordinates, we obtain the formula in

av a

E e(x) axi aij i,j=1 Definition 9. A Newtonian system with potential energy is a triple (M, g, V)

consisting of a Riemannian metric g and a potential energy V : M' -+ R. The corresponding force vector field is

X = S-grad(V). A motion in a Newtonian system with potential energy is defined to be a Mm whose tangential curve (a, b) -+ TM'" is an curve 7 : (a, b) integral curve of X.

Definition 10. The energy of a Newtonian system (Mm, g, V) with potential energy is the sum of the kinetic and the potential energy,

E:TM' R, E=T+Voir.

7. Symplectic Geometry and Mechanics

256

Theorem 15 (Energy Conservation for Newtonian Systems). Let (Mm, g, V) be a Newtonian system with potential energy, and let X = S - grad(V) be the force vector field. Then dE(X) = 0.

In particular, E(y(t)) is constant for every motion-y(t) of the system.

Proof. In local coordinates, the energy E and the force vector field are given by the formulas I M

E = 9 E 9ij(x)xY +V(x), i,j=1

X =

i=1

x' a ii -

k=1

I'v (x) +

8i 9ikW I i=1

82k

.

Using the expression for the Christoffel symbols riki from §5.7, 2

gkQ (x) (Lgii (x) + 8x (x)

-ij axQ

(X))

we obtain dE(X) = X (E) = 0 by an elementary calculation.

,

0

Motions with large energy in a Newtonian system (M'", g, V) with potential energy are-up to a change of parametrization-geodesic lines with respect to a new Riemannian metric. This construction will lead to the MaupertuisJacobi principle. Suppose that the potential energy V : Mm - R is bounded from above by Eo,

sup{V(x): xEMm} < Eo. Then g' = (Eo-V) -g is a Riemannian metric on the manifold Mm. Consider a motion y : (a, b) - MI of the Newtonian system (MI, g, V) with energy Eo,

Eo = 29('Y(t),'Y(t))+V(7(t)) Since Eo > V(y(t)), the tangent vector y(t) vanishes nowhere, and the function

s(t) := f r' (Eo - V(7(µ)))dp a

becomes a strictly monotone function a : (a, b) - (0, b* ). We invert this function and thus view t E (a, b) as a function of the parameter s E (0, b*), t = t(s). Let the curve .y*(s) be the initial curve y in this new parametrization.

7.5. Formulations of Mechanics

257

Theorem 16 (Maupertuis-Jacobi principle). Let -y(t) be a motion of the Newtonian system (Mm, g, V) with energy E0. Then ry' (s) is a geodesic line

in Mm with respect to the Riemannian metric g' = (Eo - V) g.

Proof. The Christoffel symbols1) r and 't V of the metrics g and g' are, in local coordinates, related by the formula k

rtj

=

1

k

_ 8V

1

8V

r'3 + 2 (Eo - V)

ajk

C7xi

8V ak

-&k +

02-a

9 9ij

We write the motion -y(t) _ (x' (t), ... , x'(t)) in local coordinates. Then dxk

_

ds d2xk

dsk dt

_

dt

_

- V) dt

dxk m 8V dxa 2(Eo - V)3 dt Ox- dt '

d2xk

1

1

1

2(Eo - V)2 dt2

ds2

dxk

1

1

72= (Eo

Ws-

and, using the equation of motion, d2xk dt2

--

m

rk Ix' dx'

m

ij dt i,j=1

OV ka

dt - a=1 8xa 9

as well as the energy condition m

dxi dxj

E i,j=i gij dt

dt

2(Eo - V),

we obtain the claimed result: d2xk WS-2 +

k dx' dx'

m

I'i ds ds ij=1 1

+ `1(Ec, __V) 3

m

a=1

=

8V ka 8xa9

8V

1

ka

-2(Eo - V)2 a=1 8xa9 in

dx' dx-I

ij=1 go dt dt

= 0.

Formulation of Mechanics According to Lagrange The transition to Lagrangian mechanics proceeds by considering the Lagrange function of a Newtonian system with potential energy.

Definition 11. Let (M'", g, V) be a Newtonian system with potential energy. The Lagrange function (or Lagrangian) L : TM' R is the difference of kinetic and potential energy,

L = T-Voir.

7. Symplectic Geometry and Mechanics

258

Theorem 17 (d'Alembert-Lagrange). A curve 7 : (a, b) -+ Mm is a motion of the Newtonian system (Mm, g, V) with potential energy if and only if the Euler-Lagrange equations hold: dt

(ex (y(t))) = 8x (7(t))

Proof. We prove this theorem again using local coordinates. The curve 7(t) _ (x1(t), .. . , xm (t)) is a motion of the Newtonian system with potential energy if and only if it solves the system of differential equations

ik

=-

»i

m

I

ji`ij

ij=1

- a=1 E

Va9ak

For brevity, we denoted by V. the partial derivative of the potential function, 4G, := 8V/8x°. The Lagrange function is

L('y) =

2

gijii2j - V(1'), i,7=1

and this leads to the difference d dt

8L

m

8L

(8zi

a,;3=1

d9ia

1 99. Q

ax-*l

2

axi)

axa +

m

9iaya+ V; . a=1

Multiplying the Euler-Lagrange equations by gik and summing over the index i, this system of equations turns out to be equivalent to m m a9ia 109.e M 0 = xk + M V 'k +

E i-l

i9

E ` 8x3 i=1 a.3= 1

2 axi

9

The claim now immediately follows from the formula for the Christoffel symbols r 13.. o Thus the equations of motion of Newtonian mechanics are, in the case of a potential force, equivalent to the Euler-Lagrange equations. For the latter, it does not matter that the Lagrange function arises as the difference of a kinetic and a potential energy. Hence we define:

Definition 12. An autonomous Lagrangian system is a pair (Mm, L) con-

sisting of a manifold Al" and a smooth function L : TMm - R. A Lagrangian motion is a curve -y : (a, b) -- M" which solves the system of Euler-Lagrange equations Wt

(er ('i (t))) = dL ('Y(t))

259

7.5. Formulations of Mechanics

Example 11. Let (M'", g) be a pseudo-Riemannian manifold and A a 1form on it. The Lagrange function

L(v) =

g(v, v) - A(v)

2 generalizes, in a sense to be discussed in Chapter 9, the motion of a charged particle of mass m under the influence of the Lorentz force of the electromagnetic field generated by A. Motions in a Lagrangian system can be understood from the point of view that they are critical points of the action integral L. This measures for a curve -y : [a. b] -p Mm the mean value of the Lagrange function on this curve, b

L(y) :=

Ja

Theorem 18 (Principle of Least Action). A curve -y

:

[a. b]

Mm is a

motion of the Lagrangian system (Mm, L) if and only if the variation of the action integral vanishes for every variation y1, of the curve with fixed initail and end points, y1,(a) = y(a), -y. (b) = y(b): d dµ

(jb)

0. LO =

Proof. We compute the derivative of the action integral with respect to the parameter p in coordinates -t,, (t) = (x1(µ, t), ... , x'"(µ, t)) by partial integration: =Jab

d (,C(-t,.)) 1,0

[aL((t))

- d (a (7(t)))

(0, t)dt.

The functions 8x'(0, t)/8µ are arbitrary functions vanishing at the end points of the interval [a, b], and hence the Euler-Lagrange equations are equivalent to the condition dµ (L(yµ)) Iµ=o = 0.

For general Lagrangian systems, there exists a notion of energy which, on the one hand, generalizes the energy of a Newtonian system with potential energy, and is, on the other hand, a preserved quantity. This Lagrangian energy is obtained by first introducing the Legendre transformation. Let

L : TM'

R be a Lagrange function, and let v E .,Mm be a vector at the point x E Mm. Now restrict L to the tangent space .,Aim and consider the differential D(LIT=SIm)(v) at the point v E TTMm. This is a linear map TTll1"' IR. hence a covector in T *M'.

7. Symplectic Geometry and Mechanics

260

Definition 13. The Legendre transformation C : TM'

T'M' of an

arbitrary Lagrangian system is the map ,C(v) := D(LIT=Mm)(v).

Example 12. If the Lagrange function L = Zg - V is the difference of a

kinetic and a potential energy, the following relation holds for v, w E TM':

G(v)(w) = 2 . D(g)(v)(w) = &,w). Hence the Legendre transformation L : TM"' T* M' is simply the identification of the tangent bundle with the cotangent bundle via the metric.

Definition 14. The energy of a Lagrangian system (MI, L) is the function E : TM' -' R on the tangent bundle defined by

E(v) = L(v)(v) - L(v). In the case of a Newtonian system with potential energy, we have

E(v) = L(v)(v) - L(v)

2g(v, v) + V(r(v)) .

This shows that the energy in the sense of Lagrangian mechanics coincides with the Newtonian energy.

Example 13. We compute the Legendre transformation for the Lagrange function from Example 11. Let (x, y) E TTM" with local coordinates

xl...

, xm

and y', ... , y'. Then, m

...,y"') = 2

m Egjjy'1!r - EAiy', i

1,3

and, as an element of T,M, its differential is m

m

D(LITTMm)(yl,...,ym) = mEgify'dhe -EAidy'.

i,j

i

We evaluate this map at (x, v), v = (v', ... , v'"): G(y)(v) = D(LI T:MO1) (y)(v) = m . g(y, v) - A(v) . By definition, this yields for the energy the value

E(v) = C(v)(v) - L(v) =

2 g(v, v),

which has the remarkable property of not depending on A. A more careful physical analysis shows that the energy of a charged particle indeed only

depends on the electric, not on the magnetic field. This is related to the fact that the magnetic field does no work on the particle (see §9.5).

7.5. Formulations of Mechanics

261

Remark. Denoting the coordinates on TM' by {x', , X-, x', ...'e } and the coordinates on T` Mm by {qj, ... , qm. pi, .... p,"}. we see that the Legendre transformation f- is given by

qi = x'

and

aL pi =5p,

and the expression for the energy E takes the following form:

E(x,

_ "' OL xi - L(x, x) x) - ==Y 8ii

.

Theorem 19 (Conservation of Energy for Lagrangian Systems). The energy E(y' (t)) of each motion ti(t) of a Lagrangian system (Mm, L) is constant.

Proof. The energy of a curve is

E(7(t)) _ t-1

8L

dxi

f72i

dt

- L(ti(t))

and by differentiation we obtain "'

d

dt E(Y(t))

_

ij=1

8L

dx' dx-

8L

dx'

1

C 8ii8xj dt dt + C7x'&i dt dtZ J -

' &L dx' i=Y

dxi dt

Using the Euler-Lagrange equation m

dx' "` c7L OL d2xi EY Ox-. = O±'Oxi dt + axiaxj dt2 i,7=Y ij= i3L

we immediately obtain the assertion.

0

Thus the energy is a first integral for any motion in a Lagrangian system. As in §7.3. further first integrals can be derived from symmetries of the Lagrange function. To this end, consider a one-parameter group of diffeomorphisms ,P6 : Al' M' of the configuration space as well as its generating vector field, d

V(x) := d('ts(x))j8=0 on Al'. The differentials d(4 Q : TM' TM"' are diffeomorphisms of the phase space TM"' into itself.

Theorem 20 (Noether's Theorem). Let the Lagrange function L be invari-

ant under the action of a one-parameter group of diffeomorphisms, L(d(4s)(v)) = L(v). Then the function fv : TM' --+ R defined by

fv(w) = lim

µ is a constant of motion for the Lagrangian system (Mm, L). µ-.O

7. Symplectic Geometry and Mechanics

262

Proof. Let the one-parameter group of diffeomorphisms $t be determined in coordinates by

4tt(xil ...,xm) = ('Ft(x1, ...,X"), .... Dm(x'....,xm The invariance of the Lagrange function implies that m

m aL a(De

ax= as

I

aL O4

dx'

+ E axj . ax=as

and the vector field V has the components m

V=E =1

dt

= 0,

a

s

s=IUC7-

x

a3

'

i

Thus we obtain

fv(7(t)) = 1 8x as Is.o' and from the Euler-Lagrange equation we conclude that d

d

m

m IL 81' ax= WS=0 I8=0

+

aL 02V ax' ax_as o

dxj

dt = o. o

Example 14. To each transformation group'Ft : Mm - Mm preserving the metric g and the potential function V there corresponds a first integral,

fv(w) = g(w,V), which is linear in every fiber of the tangent bundle Tlblt. We made use of this first integral in Theorem 37, Chapter 5, to integrate the geodesic flow on surfaces of revolution (Clairaut's theorem). Hence first integrals of the geodesic flow which are linear in the fibers arise from isometries of the metric.

Example 15. If 't preserves the pseudo-Riemannian metric g as well as the 1-form A from Example 11, then the Lagrangian of a charged particle in an electromagnetic field is invariant, and hence Noether's theorem can be applied to yield the invariant

fv(w) = m g(w, V) - A(V).

Formulation of Mechanics According to Hamilton We will arrive at the formulation of mechanics according to Hamilton by starting from a Lagrangian system (M, L) with bijective Legendre transformation C : TM' T'Mm. These Lagrangian systems are called hyperregular. A regular Lagrangian system is one whose Legendre transformation is locally invertible. For simplicity, we confine ourselves here to the case

7.5. Formulations of Mechanics

263

of a globally invertible Legendre transformation. For example, Newtonian systems with potential energy or the system of a charged particle moving in an electromagnetic field (Example 11) are always hyper-regular.

Definition 15. Let (Mm, L) be a hyper-regular Lagrangian system with Legendre transformation C : TM' --+ T* M' and energy E : TM' --+ R. The function

H := EoL

1

defined on the cotangent bundle is called the Hamilton function (or Hamiltonian) of the system.

Example 16. In the case of a Newtonian system (Mm, g, V) with potential energy, the Legendre transformation is determined by the formulas m

qi =

(9L and pi= axi = E gi.±

xi

a=1

Inverting this transformation leads to m

x' = qi

and

i' = E gtnPa , U=1

and the formulas for the energy E, the Lagrange function L, and the Hamilton function H read as follows:

(1) E = 2

i j=1

gijx'i +V(xl, ...,x'"),

m

(2) L = 2 E gijz'xi - V(xl, ...,xm), ij=1

(3) H = 2 E g''rpipj+V(gl,...,gm). ij=1 Through this change of phase space-i. e., replacing the tangent bundle TM"' by the cotangent bundle T'M'-we enter the realm of symplectic geometry, since T*M'" always carries the symplectic form w = d9.

Theorem 21 (Hamilton's Theorem). Let (M'", L) be a hyper-regular Lagrangian system. A curve -y : (a, b) -' AM"' is a Lagrangian motion if and only if the curve G(ry) : (a, b) -. T' M"' is an integral curve of the symplectic gradient s-grad(H) of the Hamilton function.

Proof. The Legendre transformation C is defined by i

OL

qi = x, Pi = axi . For its inverse map, we introduce the notation

x' = qi

and

i = I (ql, ... , qm, Pi, ... , pm).

7. Symplectic Geometry and Mechanics

264

The Hamilton function is then m

H= a=1

and we compute its partial derivatives:

+

1: (P. -

apt OH

a

M

a=l

E51-aL aV _ m

a

aqt

_ OL axt

OL

axt =

.

(i+)

Thus we obtain a formula for the symplectic gradient:

s-grad(H) = E t=l

A curve

G(?'(t)) = 1 21(t), ... , 2m(t), 'oil (fi(t)),

--

,

OL

is an integral curve of the vector field s-grad(h) if and only if

it = V and

Wt

(5p IL

(ti(t))) = axt ('Y(t))

The first equation is trivialy satisfied by setting it = fit, and this proves the 0 assertion.

Exercises 1. In R4 with the symplectic structure w = dxl Adx3 +dx2 Adx4. we choose the following four diffeomorphisms a, b, c, d:

a(xl,x2,x3,x4) = (21,x2+1,x3,x4), b(xl, x2, x3, x4) = (xl, x2, x3, x4 + 1), C(xl, x2, x3, x4) = (xl + 1,x2, x3, x4) , d(xl, x2, x3, x4) = (x1, 22 + 24,x3 + 1,x4 ),

and denote by r the group of motions of R4 generated by them. Prove that w is a F-invariant 2-form. Hence w induces a symplectic form on the manifold M4 = R4/F. Prove that M4 is compact. Finally, denote by [F, F) the commutator group of F. Then F/[t, rJ is a free abelian group of rank three (Thurston, 1971).

Exercises

265

2. Prove that the symplectic form w of a compact symplectic manifold M2m can never be an exact differential form. Hence the second de Rham cohomology HDR(M2m) is non-trivial. In particular, for m 1, even-dimensional

spheres St' have no symplectic structure.

3. Consider M = R2\{0} with the symplectic form w = dx n dy and the vector field

V_

x

8

y

8

x2+y2 8x+x2+y2 8y

a) Let cp = arctan(y/x) be the polar angle defined in every sufficiently small neighborhood of (x, y)

(0, 0). Compute grad(W) and

b) Conclude from this that V is no Hamiltonian vector field on all of M.

4 (Continuation of Example 3). Prove that the Liouville form on the twodimensional coadjoint orbit through the element (a, Q) E g` (3 54 0) is given by the expression

w=

daAdf

.

Hint: Show first that the fundamental vector fields corresponding to the Lie algebra elements (1, 0) and (0, 1) are

aa, and - Qom. 5. Let V be a vector field on a manifold Mm and 1 = V(x) the associated differential equation. A first integral of V is a smooth function h : Mm - R which is constant along every solution of the differential equation. a) Prove that h : Mm -p JR is a first integral of V if and only if dh(V) = 0.

b) Prove that the set Cv (M'n; R) of all first integrals of V is a subring of

C' (M'; R). c) Show that h(x, y) = y2 - 2x2 + x9 is a first integral of the vector field

V = 2y 8/8x + (4x - 4x3) 8/ay in R2. Describe the geometric shape of the integral curves in the (x, y)-plane by means of h.

d) Find a vector field in the plane which has no non-trivial first integral.

6. Let f = a + b i : U -p C be a holomorphic function on an open subset U of JR2 ^_' C. Consider the vector field V = grad (a).

a) Prove that b : U -. R is a first integral for V.

266

7. Symplectic Geometry and Mechanics

b) Describe the geometric shape of the integral curves of this vector field for the functions f(z) = zk and f(z) = z + 1/z. 7. Consider on ]R3 a vector field B (a magnetic field) and its 2-form

B= as well as the symplectic form on the phase space R3 x JR3 with coordinates (9, ii) = (x, y, z, Vx, Vy, Vz),

wB = m(dxAdvx+dyAdvy+dzAdvz) - - B. C

As Hamilton function, we choose the kinetic energy

Ho = 2 (vZ+vy2+vZ). a) Show that the defining equation for the Hamiltonian vector field associated with Ho is equivalent to the Lorentz equation dv

_

e

c.vxB.

mdt

(*)

b) If B = dA is exact and A denotes the associated vector field, show that the map

f:

1R3 x JR3 -R 3 x 1R3,

(4> ) ' (q, mii + A)

(q, p)

is a canonical transformation, i.e., for the canonical symplectic form wo and the Hamilton function HB,

wo = dxndpx+dyAdpy+dzndpzi HB =

-lip-

-All

the following equations hold:

f*wo = 'B, f*HB = Ho, and equation (*) does not change.

c) If B is constant, any particle moves on a helix. Hint: Interpret torsion and curvature as first integrals. If both these quantities are constant for a curve, then this curve has to be a helix (Exercise 6 in Chapter 5). 8 (Plane Toda Lattice). Consider in JR2 the second order differential equations

i=

-ex-y,

y=

ex-y

which in phase space (x, y, i, y) E JR4 are equivalent to t; = X

a a

X(x,y,i,y) = a TX + y ay -

ex-y a +

ai

The energy E = (i2 + y2)/2 + ex-y is a first integral.

ex-y

ay

with

267

Exercises

a) Show that this system has further first integrals, for example P = i + y or K = (i - 2y)(y - 2i)/9 - e---y. These quantities are related by E + K = 5P2/18.

b) The set M2 (E, P)

(x, y, i, y) E 11 P4 :

(i2 + y2)/2 + eZ-v = E, ±+y=P}

is not empty if and only if 4E - P2 > 0. In this case M2(E, P) is a smooth two-dimensional submanifold of R4 without boundary . This leads to a decomposition of ]R4 into a family of submanifolds, and every integral curve of X lies completely in one of them. c) Show that each of the submanifolds M2(E, P) lies in an affine subspace of dimension three and is diffeomorphic to R2.

d) If {(t) = (x(t), y(t), i(t), y(t)) is an integral curve of X in M2(E, P), then

x y i+y = P and f 4E-P -4ez-V =

t

Show that (A > 0) dz

A -Bez

_=

1

In

v/-A

vA- A -Bee ( I + A -Bee

'

and use this formula to integrate the equations for the integral curves in M2(E, P) completely.

9 (Euler Equations). Let I : ]R3 - R3 be a symmetric positive definite operator. Consider the differential equation I(i) = 1(w) x w, where x denotes the vector product.

a) Prove that this differential equation has two first integrals, the energy 2E = (I(w),w) and the momentum M2 = (I(m), I(w)). b) Conclude from this that the integral curves of the differential equation are intersections of two ellipsoids with center 0 E IR3 (I 0 IdR3). In particular, all integral curves are closed. Let Il be the smallest and 13 the largest eigenvalue of I. Then there exists an integral curve only for 2EIl < M2 < 2E13. . 10 (Motion in a Central Force Field). Let a potential force F act on a point of mass one in R3, dU

-grad(U(r)) F drr) r with a function U(r) depending on the radius r = IIxil only. The energy E = 11±112/2 + U(r) is a first integral.

7. Symplectic Geometry and Mechanics

268

a) Show that M = x x . is a further first integral (M is the angular momentum), and conclude from this that every trajectory of the point lies in a plane in R3. b) Determine for which values of the parameter M E JR3 the level surface

A3(M) = {(x,x)ER6:xx:i=M} is a three-dimensional submanifold of phase space R6.

c) Let r(t) be the distance to the origin of a motion whose an;;ular momentum is M. Then d2r

_ _dr dU

dt2

1IM112

+ _73-

i.e., r(t) describes the motion of a point in R1 under the effective force F2 = -grad(V) with effective potential V(r) = U(r) + IIMII2/2r2. The energy of this motion is

E* = 2 + U(r) +

I I2r2Iz

Prove that this energy E* coincides with the energy E (for fixed angular momentum M).

Hence, if x(t) E R3 is the trajectory of a point moving under a force F and r(t) is its distance to 0 E K3, then

V2E -

IIMII2

- 2U(r) = r',

i.e.

r

J

-dt = t

11 (Classical Moment Map). Consider K3 with the defining representation of SO(3, K).

a) Prove that this representation is equivalent to the adjoint representation of SO(3, K) on its Lie algebra so(3, K), if R3 and so(3, K) are identified via the map vl R3

so(3, f8),

v = v2 u.- v =

0

-V3

v::

v3

0 vi

-V1 0

-V2

V3

Moreover, the adjoint and the coadjoint representation of SO(3, K) are also equivalent.

b) Prove that this identification satisfies the equation

v(w) = v x w, [v, w] = [v, w],

(v, w)

Ztr (vv-) .

269

Exercises

c) By a), the moment map' : T'R3 - so(3,R)* can be interpreted as a map from T*R3 to R3. Show that it can then be written in the form

W(9,P) = 9XP, and hence it coincides with the classical angular momentum.

Chapter 8

Elements of Statistical Mechanics and Thermodynamics

8.1. Statistical States of a Hamiltonian System The Hamiltonian formulation of mechanics starts from a configuration space

X"' and makes use of the phase space T'X' with its canonical symplectic structure. A state of the mechanical system under consideration is a point in the phase space T`X'", and the motions of states are the integral curves of the symplectic gradient s-grad(H) of a Hamilton function H : T* X' --+ R. In this formulation of mechanics, the only essential data to be given are a symplectic manifold M2rn and a function H. The point of view of statistical mechanics is based on the idea that, for instance because of the size of the mechanical system or as a consequence of the imprecision of measurements, the state of the mechanical system cannot be determined precisely by fixing 2m real parameters. Instead, to each open set U C M2"', we can only ascribe the probability p(U) that the state belongs to the set U. This leads to the concept of considering mechanical states no longer as points in the phase space M2, but as probability measures on M2'".

Definition 1. Let a Hamiltonian system (M2"', w, H) consisting of a symplectic manifold and a Hamilton function be given. A statistical state is a probability measure p defined on the a-algebra of all Borel sets on M2'".

271

8. Elements of Statistical Mechanics and Thermodynamics

272

Example 1. For each point x E M2n', consider the measure 62 concentrated at this point,

a:(U)

1 0 ifx¢U, l 1 if a E U.

Therefore, classical mechanical states are particular statistical states.

Let -tt M2" -+ M2in be the flow of the symplectic gradient s-grad(H). The motion of the classical state x E MZ"' is the trajectory 0t(x). For the probability measure b+,(s) corresponding to the point it(x), we have :

t

xE (U) 1 = bx(-tt I(u)), 60&)(U) = 1 0, 1 , x E t 1(U ) and this formula leads to the following definition.

Definition 2. Let a Hamiltonian system (M2-, w, H) and a statistical state p be given. The motion of µ under the impact of the Hamiltonian system is the curve At of measures

At (U) := µ(4 1(U)) Definition 3. An equilibrium state of a Hamiltonian system (M2m,W, H) is a state A which does not change in the course of the motion of s-grad(H), At = P.

Definition 4. A statistical state p of a Hamiltonian system (M2in,W, H) has a stationary terminal distribution if the equation

'U-(U) := lim t 00 MO) defines a Borel measure on M2"'.

Theorem 1. A stationary terminal distribution µ«, is always an equilibrium state. Consider the volume form (-1)m(m-1U2

dM2m _

Wm

m!

of the symplectic manifold and the Borel measure induced from it (see the final remark at the end of §3.5). If the measure p is absolutely continuous with respect to the volume measure,

µ(U) := je(x).dM2m(x), then a is called the density function of the state µ.

8.1. Statistical States of a Hamiltonian System

273

be a state with density Theorem 2 (Liouville's Equation). Let p = function, and let at be the motion of this state in a Hamiltonian system. Then the measures pt = pt dM2" are also states with density function, and

d

dtet =

-{H,e}o$_t.

Proof. The flow ibt consists of symplectic transformations and preserves, in particular, the symplectic volume form dM2in, 4 (dM2i`) = dM2m. Transforming the integral, we see that pt = P o &_t is the density function of the state µt. This implies dt of = dt

P o D_t = do o dt

-t = -

o

-t = -{H, p} o O_t.

0 Corollary 1. A statistical state µ with density function P is an equilibrium state if and only if a is a first integral of the Hamiltonian H, (H, g} = 0. If x E M2m is a classical state in Hamiltonian mechanics, the value H(x) of the Hamilton function H is the energy E of the state,

E(x) = H(x) = JM2rn The definition of the energy of a statistical state generalizes this relation:

Definition 5. The energy of a statistical state p is the integral

E(µ) =

JMom

H(x) dp(x),

if this integral exists.

Theorem 3 (Conservation of Energy in Statistical Mechanics). If u t is the motion of a statistical state in a Hamiltonian system, the energy E(pt) is constant.

Proof. After transformation of/ the integral, we immediately obtain

E(pt) = l

H o I 1(x) dtt(x),

J M2m

and differentiating this with respect to the parameter t yields the formula

d E(µt) = fM2m {H, H}(x) du(x) = 0.

0

Now we want to study how the probability 1At(N2m) that the state µ = B dM2ri is in the compact subset N2ni C M21 at time t changes in time. The following lemma serves as a preparation.

8. Elements of Statistical Mechanics and Thermodynamics

274

Lemma 1. Let (M2i', w) be a symplectic manifold.

Then, for any two

functions f, g : lbl2in --+ R, the following formula holds:

df AdgAw,,,1 =

1 If, M

Proof. To prove this formula, we choose on 1112»' local syinplectic coordinates (ql, ..,P.),

w=

dp0 A dqq ,r= I

Setting Ai := dpi Adgi, we see that the exterior product satisfies the relations

AiAAj = AjAAi and AiAAi = 0. Since the forms Ai commute with one another, we can use the binomial formula to compute the exterior power wk, m

wn, _

(A1)'

>

1

m.

Aa' A

A A"

Because Ai A Ai = 0, only summands with rri < 1 occur in this sum, and hence it reduces to a single term:

writ = in!.AlA...AAna = m!.(dpiAdgi)A...A(dpmAdq»,) We compute WI-1 in a similar way, and obtain ",

(m-1)!

AlA...A,;...A

Thus, the following exterior products vanish:

(1) dpi Adpj Aw"'-1 = 0 = deli Adgj

Awl"-

= 0 if i # j

(2) dpi A dqj A w'

and we obtain fit

df A dg

Awrr-1 =

(.!m- dg dp, A dqi + 2f ag dq, Adpi) api Oqi OCli dpi rn

('m - 1 ) !

Of yg pi

arr.-1 mi

.

Tqi-

Of 09

di

w"'

A I A ... A A,,,

CdR

{f,g} wr

The relations m. d(f dg A w'"-1) = in df A dg A w"'- I = {f, gj w"' together with Stokes' theorem imply, after integration.

8.1. Statistical States of a Hamiltonian System

275

Theorem 4. Let (A12mw) be a compact symplectic manifold with boundary. Then

J12in

f.dg nwm-1

{f,g} w"r i)At 2°

.

In particular, the integral

J r2" {f,9} w"` = 0 vanishes for any compact symplectic manifold without boundary.

Corollary 2. If the Poisson bracket if, g} of two functions defined on a compact symplectic manifold 1112" without boundary does not change sign, then it vanishes identically, If, g} = 0.

We will apply these formulas to a statistical state with density function e.

Definition 6. The (2m. - 1)-form

At)

A'dHAw"'-1

(m-1)!

is called the probability current of the statistical state it = e dM2"' in the symplectic manifold (Al2m,w) with Hamilton function H.

Theorem 5. Let lit be the motion of the statistical state p = e dA12"' in the Hamiltonian system (1112"', W, H), and let N2", C AI2si be a compact submanifold. Then (µr(N2-)) (r=o

Wt

= JNFIJ

Proof. We compute the derivative with respect to time at t = 0 and use Lionville's equation as well as the preceding integral formulas: d N2", {He} dM2"r dt r=e dM2", o-

_J

J

N2m

N2s'

(m - 1)!

J

e dH n w"'-i =

UN2"4

f

j(Ic) .

aN2"

The probability current j(µ) is a hypersurface measure on all (2m - 1)dimensional submanifolds of the symplectic manifold M21. It expresses the infinitesimal change of the probability for the state it to be in the set N2i' C A12"' as an integral over the boundary of N2'".

Example 2. If the subinanifold N2i" C M2" is described by inequalities of the Hamilton function,

N2n = {x E M2" : C1 < H(x) < C21,

8. Elements of Statistical Mechanics and Thermodynamics

276

then the differential dH vanishes on the tangent bundle T(8N2m) of the boundary, and so does the form j(p). We hence obtain, for the motion At of every statistical state p = p dM2,, dtpt({x E M2in :

C1 < H(x) < C2}) = 0.

Thus the probability for a state p to be in N2in at time t is constant.

Example 3. In case m = 1, the formulas for the probability current of a two-dimensional Hamiltonian system (M2, w, H) simplify:

j(p) = e dH and dpt(N2) = JaN2 p dH. t Now we turn to the notion of information entropy for a statistical state. As a motivation, we first recall the notion of information of an event introduced by C. E. Shannon (1948). The heuristics is the following: If, in a series of experiments, an event occurs with probability close to one, then the "information" contents of this event is small. Conversely, if the probability of this event is close to zero, the occurrence of this event contains a large amount of "information". Modeling the probability computation by a triple (i, 21, p) consisting of a set i, a or-algebra 21 of subsets of S2, and a measure p defined on 21 such that A(fl) = 1, we arrive at the following definition for the Shannon information.

Definition 7. Let (S2, 21,,u) be a probability space. The amount of information contained in an event A E 21 is

1(A) := - log(p(A)). For a finite set S2, one forms the mean of these information amounts and gets in this way the so-called information entropy.

Definition 8. Let (12, p) be a finite probability space. The information entropy of this space is S(Q, p)

f

t

1(w)

. dp(w) = - E pi

log(pi) ,

i=1

where the set 11 consists of the elements {w1, ...,wn}, and pi := p({wi}) is the probability for the event wi. The information entropy is a measure for the uncertainty of the probability space. In fact, if St = {wl, ... , w } consists of n points and we denote by p' the measure corresponding to the equidistribution, p'(wi) := 1/n, then the following holds.

8.1. Statistical States of a Hamiltonian System

277

Theorem 6. The information entropy of a finite probability space (S),µ) does not exceed the information entropy of the equidistribution: S(Q, Ft)

<_ S(n, µ')

Equality occurs if and only if u = µ' is the equidistribution.

Proof. The proof is based on the fact that the function f (x) := x log(x) is convex on the half-line (0, oo), since

f'(x) = 1 + log(x) and

f"(x) = x > 0.

The convexity of f then implies the inequality f (pi)

n

f

pi

<_

i=1

n

n

= n E pi i=1

log(pi )

i=1

n

pi = 1, one immediately deduces from it that

Using the relation i=1

> pi log(pi) < -log (n) =

S(1, lt)

S(S2,

i=1

The formula S(12,,u) = - E pi log(pi) leads to the notion of information entropy for a statistical state with density function.

Definition 9. Let (M2,, w, H) be a Hamiltonian system, and let a = e dM2m be a statistical state with density function. We define its information entropy as the integral

S(u)

- JM2m P log(P) dM2m,

if this integral exists.

Remark. This notion of entropy is not without problems in statistical physics. The following result concerning the conservation of entropy claims that the information entropy does not change in the course of the motion of a state. This is a good assumption for reversible processes (and processes which can be approximated by reversible ones), but cannot be applied to the vast range of irreversible processes, in which the entropy increases. For

this reason, we will apply the notion of information entropy S(µ) only to equilibrium states. Theorem 7 (Conservation of Information Entropy). The information entropy S(pt) of a statistical state with density function is constant under the motion of a Hamiltonian system.

8. Elements of Statistical Mechanics and Thermodynamics

278

Proof. We differentiate the function S(Pt) with respect to the parameter t and use Liouville's equation: dt S(At)

JM2m

(d-tL-log(owL

dM2-

({HQ}l(Q)+{HQ}) dM2-.

_ M2m

But since {H, log(e)} = {H, a}/e, we have

{H,e}+{H,g}.log(e), and this yields dt S(µt)It_o =

JM2m{H, e

log(e)} dM2m

(-1)m(m-1)/2

m

J

M

(-1)m(m-1)/2 ( m-1 )

JMs"`

dA/ d(e log(e)) A

w"`-1

Since M2m has no boundary, Stokes' theorem immediately implies the assertion.

Remark. In the case of a manifold M2"' with boundary we obtain (-1)m(m-1)/2 d dt

S({pt)It

IOM2.

(

e log(e) dH n

w, H) has several special equilibrium states. These are characterized as realizing the maximum of information entropy in a particular class of statistical states. We will discuss two of these states, the so-called Gibbs state (or the canonical ensemble) and the microcanonical ensemble. We will, in general, make the following three assumptions for the Hamiltonian system (M2m, w, H):

Assumption 1. The Hamilton function H is non-negative,

H(x) > 0

for all x E M2s` .

Assumption 2. For every positive number B > 0, the following integral exists:

Z(B) :=

r

e-N(x)/B. dM2m(x)

JM

The function Z(B) is called the partition function of the Hamiltonian system.

8.1. Statistical States of a Hamiltonian System

279

Assumption 3. For every positive number B > 0, the following integral exists: H(x) . e-H(=)lB . dM2m(W ) . bl2m

Remark. The parameter B will later be identified with absolute temperature (multiplied by the Boltzmann constant k). Its appearance here indicates the exceptional role that temperature plays among all thermodynamic parameters. The function

E(B) :=

I

H(z)e-H(:)/B , dA12-(z)

Z(B) M2m is called the inner energy of the Hamiltonian system (M2m, w, H); its inner entropy is E(B) S(B) := log(Z(B)) + B Finally, the free energy F(B) is defined by

F(B) := -B log(Z(B)) = E(B) - B - S(B). First we note that the inner energy E(B) is a non-decreasing function on the interval (0, oo). Its derivative is

dE _ TB

1

Z2(B) B

[(I

M2m

e-

dM2m(x)) (ML H2( l2

_

- (ML H(x )e

z)e-dM2m(x))

dM2m(x))

and the Cauchy-Schwarz inequality shows that this derivative is positive

(H j4 const). Denote by En and Em., respectively, the bounds of the range of the inner energy E : (0, oo) - R. We calculate the derivative of the partition function in a similar way: dZ

1

dB = B2

JM2m

BZ Z(B) E(B).

Hence the inner energy as well as the inner entropy can be expressed in terms of the partition function. We summarize these formulas. Theorem 8 (Simple Equation of State for a Hamiltonian System).

(1) E(B) = B2gR (log(Z(B)));

(2) S(B) = W(B log(Z(B)));

8. Elements of Statistical Mechanics and Thermodynamics

280

(3) in the sense of 1 forms on the one-dimensional manifold (0, oo),

dS = B dE. Proof. The third equation is a straightforward consequence of the first two.

a The Gibbs state (canonical ensemble) is distinguished by the property that it realizes the maximum of the information entropy among all stag of fixed energy.

Theorem 9 (Gibbs State, Canonical Ensemble). Let an energy value Eo between the minimum and the maximum of the energy function E(B) for the Hamiltonian system (M2m, w, H) be given. Among all statistical states

p = P dM2' with density function of energy E(p) = Eo, there exists precisely one state, PGibbs = PGibbs dM2m, of maximal information entropy

S(pcibb.). The density function of this state is 1 -H(x)/Bo , PGibbs(x) = Z(Bo)e

where the parameter Bo is determined as a solution of the equation E(Bo) _ Eo. The value of the maximal entropy S(pCibbs) is

S(pGibbs) = S(Bo) = log(Z(Bo)) +

E(BO) o)

Proof. Choose Bo as a solution of the equation E(Bo) = E0, and )et e-H(x)/Bo dM2m 1 Z(Bo) be the corresponding Gibbs state. Its inner energy is pGibbs =

E(pcibb.) = Z(B0)

H(x)e-H(-)/Bo

JM2m

. dM2m = E(Bo) = Eo,

and hence UGibbe is a statistical state of energy Eo. For every oti er state p = P dM2m with energy

E(IL) = fM2m H(x)P(x) dM2'n = E0, we consider the function

f (t) = S((i - t)pGibbs + tp)

,

the information entropy of the statistical state (1 - t)pGibbs + tp. Differentiating f (t) with respect to the parameter t, we obtain

d2f(t) dt2

- _ JU2-

(e - PGib.)2 (1 - t)PGib, + tP

dM2' <

0.

8.1. Statistical States of a Hamiltonian System

281

Moreover,

df = dM2m - S(PGibbs), dt y2" e - log(eGibbs) and calculating the integral using E(p) = Eo leads to

f

J

M2"'

e log(gGibbB) - dM2"'

=

log(Z(Bo))) dM Jim g( go -

_ -log(Z(Bo)) - BoEo = -S(Bo) We conclude that the derivative of f vanishes at t = 0. Hence the derivative is non-positive, and f (t) is decreasing in [0, 1]. Since f (0) > f (1), we obtain

S(P) <

S(lpGibbs)

-

If S(µ) = S(PGibb.), the first derivative of f (t) vanishes identically. Hence the second derivative of this function is also zero, and from the formula stated before we conclude that e = eGibbs The assertions of the theorem 0 are proved.

Example 4 (Maxwell and Boltzmann Distribution). In classical statistics, the Hamilton function of a particle with potential energy depending only on the coordinates q can be written as 2

2

2

H(q, p) = 2m + U(q) =

P1

+2m+ 3 + U(q).

Thus, the partition function is equal to

Z(B) = I exp [_i?

] dpiAdp2ndp3

which, using the integral fR a-QS2dx =

f

r

exp I

qJ1

L

r/a, can also be written as

Z(B) = (27rmB)3/2 - ZZ(B) ;

Zq(B) denotes the value of the integral in which the unknown potential energy occurs. In the same way, the Gibbs state factors into the probability measure Pp of momentum values and that of coordinate values Pq, PGibbs = lip 'A q =

(27rmB)3/2e-p2/2mBdP

Zqe-U/Bdq

Going from the momenta to the velocities, we arrive at the so-called Maxwell distribution, 3/2exp [m(vl +v3)] dv1dv2dv3, Pp 2B 2M ) which again is the product of probability distributions for every single velocity component. The Boltzmann ensemble is defined to be the probability

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282

measure p. and is proportional to the particle counting measure v. For example, for the potential U(x, y, z) = ingz of a. homogeneous gravitational field which is parallel to the z-axis, this yields the barometric formula V = 1/0 t,-inyz/'idx. d y dz.

The second natural equilibrium distribution of a Hamiltonian systc n starts from a subset A C A12ni of the symplectic manifold which is invariant under the flow 4Dt of the Hamilton function H and has positive finite volme, 4bt(A) C A,

0 < vol(A) < oc.

Consider the density function

j 1/vol(A) if x E A, `°A (x) = l if .r E A, 0 012m. This is an equilibrium state, as well as the statistical state ILA = eA since A is invariant under the flow 4bt. Its information entropy is

S(,A) = - f

ll,PA(x) - log(PAW) d12i'(x) = log(vol(A)).

Theorem 10 (Microcanonical Ensemble). Let A C M2"' be an invariant subset for the flow of the Hamiltonian system (A12", w, H) with positive finite measure. If it = p dM2"' is a statistical state whose density function has support in A, supp(p) C A. then its information entropy is always bounded from above by the information entropy of p.A,

S(Ez) < S(tA) . Equality only occurs for it = AA.

Hence the microcanonical ensemble IAA realizes the maximum of the information entropy among all statistical states concentrated in the set A.

Proof. The proof proceeds in complete analogy to that of the preceding theorem. Again, we consider the function f(t) = S((1 - t)IA 4 tit) and calculate its second derivative:

d2f dt2

=

(e - UA)2

. dA12,n < 0. -(1-t)eA+p fA

The derivative of f at t = 0 vanishes: df I

dt t_tt

=-

f

A

(e-

-log(eA) f

since It is a probability measure with support in A.

0,

A

0

8.2. Thermodynamical Systems in Equilibrium

283

8.2. Thermodynamical Systems in Equilibrium First we generalize the simple equations of state of a Hamiltonian system by keeping a fixed symplectic manifold (M2",, w) but considering a family of

Hamilton functions H(x, y) depending on an additional parameter y E N'. Let the parameter space also be a smooth manifold, and suppose that the Hamilton function H(x, yo) satisfies the three assumptions of the preceding section for any fixed value of the parameter yo E NT. Then the partition function Z(B, y), the inner energy E(B, y), the inner entropy S(B, y), and the free energy F(B, y) become functions on the manifold (0, oo) x Nk,

Z,E,S,F: (0,oo) x Nk -4 R, and, of course, the equation of state B dBS = dBE holds for each fixed value of the parameter yo E The general equation of state of a Hainiltonian system depending on parameters is a relation between the complete differentials considered as 1-forms on (0, oo) x N".

Theorem 11 (General Equation of State for Parameter-Dependent Hamiltonian Systems). In the sense of an equality for I-forms defined on the manifold (0, oo) x N", one has

dE = B Proof. By the definition of inner energy, we have

B

dS + S dB = log(Z) dB + B d(log(Z)) + dE,

and again inserting S, yields

dE. Remark. If we allow the until now independent variable B to be a function

B : N" -> (0, oo) on the parameter space N", we may consider all the functions Z, E, S and F as functions on N' by means of the map

N'

(0, oo) x N",

y -' (B(y), y).

The exterior derivative commutes with the pullback of forms and thus implies on the manifold N" the equation

dE = B dS + E dB - B d(log(Z)).

284

8. Elements of Statistical Mechanics and Thermodynamics

The macroscopic description of thermodynamic systems starts from a few assumptions. First, all thermodynamic systems should be in mechanical equilibrium. Furthermore, it is supposed that no particle exchange takes place during the change of states. Lastly, the macroscopic state of the thermodynamic system is to be completely described by a parameter y E Nk, which is an element of a finite-dimensional manifold. If this holds, the following phenomenologically observed laws are historically known as the fundamental theorems of thermodynamics. In a modern account of theoretical thermodynamics, they are taken as axioms:

Zeroth Fundamental Theorem of Thermodynamics. There exists a positive function T : N'' - IR, called the temperature.

First Fundamental Theorem of Thermodynamics. There exist two (not necessarily exact) 1-forms dA and dQ on the manifold Nr such that their sum

dA + dQ = dE is an exact form. The function E is called the energy of the thermodynamic system, dA is called the work form, and dQ is called the heat form.

Second Fundamental Theorem of Thermodynamics. The 1 -form dQ = dS

T

is an exact form, and the resulting function S is called the inner entropy of the thermodynamic system.

At this level of abstraction, a thermodynamic system in its macroscopic description is a 4-tuple (NT, T, dA, dQ) consisting of a parameter manifold

N'', a function T : N' -+ R, as well as two 1-forms dA and dQ such that the 1-forms

dA + dQ and dQ T are exact. Of course, the following relation holds:

dE = dA + T dS , and this is Gibbs' fundamental equation of thermodynamics. Now, with each parameter-dependent Hamiltonian system a thermodynamic system can be associated in a canonical way. In fact, let a symplectic manifold (M2n', w) and an additional manifold N'' be given. Fix two non-negative

8.2. Thermodynamical Systems in Equilibrium

285

functions,

B : N'

and H : M2m x N' --10, oo) ,

(0, oo)

and consider H as a family of Hamilton functions.on the symplectic manifold.

Using the Gibbs state, e-H(z.b)lB(b) . dM21n(x) = we define the function T : N' --+ R1 and the 1-forms dA and dQ on N' as

pGibt)s

follows:

(1) The temperature of the thermodynamic system is the function B divided by the Boltzmann constant k (= 1.380.10-23W/K):

T(y) := B(y) k (2) The energy of the thermodynamic system is the inner energy of the Gibbs state uGG ib) bbs'

E(y) = E(B(y)) =

Z(B(y)) M2'"

H(x,

y)e-H(x,b)/B(b)

. dM2"'(x) .

(3) The entropy of the thermodynamic system is the inner entropy of the Gibbs stateuB(b) Gibbs*

S(y) = k S(iz Y)

= k - log(Z(B(y))) + k E(B(y)) B(y) (4) The heat form dQ is the product of temperature with the differential of entropy,

dQ := (5) The work form dA is

dA =

E

dB - B d(log(Z)).

The First Fundamental Theorem of Thermodynamics,

dA + dQ = dE, is, in the model just described, an automatic consequence of the general equation of state for parameter-dependent Hamiltonian systems. Hence, by means of the Gibbs states, a thermodynamic system is associated with every such Hamiltonian system. Let V be a vector field defined on the manifold

N. Then

dA(V) = B

Z

8. Elements of Statistical Mechanics and Thermodynamics

286

and, since

V(-H/B)e-H/BdM2m = - JV(H)e/I5dM2m+Y-E.Z,

V(Z) = J

M2m

,%12m

we obtain an explicit formula for the work form,

Jwzm V(H) - e-B dM2m .

dA(V) =

Z The pressure pi, of the thermodynamic system generated by the vector field V on the parameter space Nr is the function

-

V(H(x, y)) e H(=.v)/B(y) . dM2-(x) .

I Z(B(y)) M2" Thus the relation dA(V) = - pv holds by definition, and the work form PV (Y)

can be represented with respect to a frame V1, ... , Vr of vector fields of the manifold Nr in the form r

dA = -

PV.

Qi,

i=1

where a,, ... , ar is the dual frame of 1-forms on Nr.

Example 5. The pressure p as used in thermodynamics is obtained if the parameter manifold N' is one-dimensional and describes only the volume V, and V = O/8V is chosen as the vector field. It is known that then the only contribution to the form dA comes from the mechanical work of volume change,

dA = -p - dV. Considering in the continuum limit for a large number of particles their num-

ber as a continuous variable, we see that the amounts of matter nl, ..., nr of a system consisting of r different chemical components can be chosen as parameters for the manifold Nr. In this case, the chemical work is traditionally expressed by the so-called chemical potentials µl, ... , µr (which the reader should not confuse with statistical states): r

pi dni.

dA = i=1

The chemical potential µi of the i-th matter component is the negative of the pressure associated with the vector field O/Oni. Similar formulas express the electromagnetic work in the presence of magnetic or electric dipol-momentum densities.

Consider, in particular, the case that there exists a vector field V on the parameter space Nr such that V(T) - 1 and V(H) - 0.

8.2. Thermodynamical Systems in Equilibrium

287

All the thermodynamic quantities are then computable starting from the partition function Z

M2m

e-H/kTdM2m

or the free energy derived from it,

F = -k T log(Z) . Theorem 12. Suppose that there exists a vector field V on the parameter space Nr such that V(T) = 1 and V(H) = 0. Then: (1)

S = -V(F) = -dF(V) = -V i dF.

(2) E = F-T-V(F) = (3) dQ = - T Lv(dF). (4) dA = (5) The pressure generated by an arbitrary vector field W is

pw = - dF(W) + dF(V) dT(W) . In particular, the vector field V generates no pressure, pv - 0.

Proof. Differentiating the partition function Z with respect to the vector field V, we obtain

E-Z He-H/AT . dM2m = k.T2 V(Z) k T2 M2m This implies the following formula for the derivative of the free energy,

J

V(F) = -k log(Z) - k T

k

T2 = -k (log(Z) +

k

T) = - S,

and the first of the assertions is proved. The relation

S = -V(F) = -(V J dF) implies

dS = -d(V J dF) = -,Cv(dF). The expression for the heat form is calculated from its definition,

dQ = - T Lv(dF) . The energy E is determined by the free energy F as well as the entropy,

E=

F-T(VJdF).

Lastly, we calculate the work form:

dA = dE - T dS = d(F - T(V J dF)) + TLv(dF) = dF - (V J dF)dT.

8. Elements of Statistical Mechanics and Thermodynamics

288

Example 6 (Ideal Gas). We view an ideal gas as n points of equal mass M in a volume region V C 1R3 which do not interact with one another. The configuration space of the mechanical system consists of the product of n copies of the set V factored by the action of the permutation group consisting of n! elements. If we want to avoid this group identification, we can simply consider the phase space M := T'V x ... x T'V equipped with a measure which already takes into account this identification: 3n

dM := i=1

The Hamilton function of the mechanical system is

i 2111

X(91,Q2,43).

Here X denotes the characteristic function of the set V C R3. Calculating the partition function of the thermodynamic system depending only on the two parameters T and V leads to the formula

Z(T V) =

je_hh1TdcAdp = m

el2MkTdp/I1

1J

`

3n

oo

n

;j (27rkTM)

3n/2

.

We apply Stirling's formula log(n!) n (log(n) - 1) and calculate all the thermodynamic quantities for the ideal gas using this approximation. The free energy is

F = - k T log(Z)

nkT (log(V/n) + 1 + 3 log(2akTM)) .

We obtain the entropy and the energy from the free energy according to Theorem 12, since the Hamilton function does not depend on the temperature:

S=-

OF

3 = nk (log(V/n) + log(T) + coast) ,

E=

OF

2 3n

O

The pressure p of the ideal gas corresponds to that of the vector field described by the volume, and is determined by

P=-

OF 8V =

n

Here n/V is the particle density of the ideal gas.

8.2. Thermodynamical Systems in Equilibrium

289

Example 7 (Solid Body). A solid body consists of n points oscillating around a point. If we take this point to be the origin, the Hamilton function is

H

n

+ 3i - ` \(p3i-2 +i-1 2Mi

2 3i-2 + q3i-1 2 + q3i 2

+ K2

2

i=1

Introducing the basic frequencies

v

Ki

2,r

Mi'

we obtain as the value of the partition function Z

(k T) 3n

1

v3

.... yn

The free energy is

F = - k T log(Z)

kT (n log(n/e) + 3n log(kT) + const) .

Starting from this, one can calculate the other thermodynamic quantities, for example the energy:

E = F-T 5T _ - T2 5T 49

(7-F,)

=

This formula says that every degree of freedom of an atom carries the mean

inner energy k T, a result known as the Dulong-Petit rule. However, at temperatures close to absolute zero, this rule no longer holds, since the model for a solid body as being a combination of harmonic oscillators is not adequate anymore.

The inner and the free energy are examples of so-called thermodynamic potentials. Classical thermodynamics usually adopts the point of view that the choice of a particular potential already determines the quantities by which the system is to be described. In this sense, the inner energy E is understood to be a function depending upon entropy and volume, the free energy F is a function of temperature and volume, etc. Consider a Hamilton function not depending on temperature. As in the two preceding examples, the free and the inner energy are related to the vector field V dual to the temperature via the relation (Theorem 12)

E=

O

Hence the inner energy E appears to be the Legendre transformation with respect to temperature of the free energy chosen as Lagrange function, L :_ -F (see §7.5). By means of Legendre transformations with respect to other parameters, one obtains the other thermodynamic potentials, enthalpy and free enthalpy, which we will not discuss in detail here. In the case that only

8. Elements of Statistical Mechanics and Thermodynamics

290

the mechanical work of the volume change contributes to the work form, the First Fundamental Theorem reads as follows:

dE = OE OE T = and = aS p -VV

If the energy is even twice continuously differentiable, the interchangeability of second partial derivatives is equivalent to a so-called thermodynamic or Maxwell relation 02E OT Op

avas = aV = OS

Beginning with the free energy,

dF = we similarly obtain the equations

S

,

p= -aV,

as well as the thermodynamic relation aS

ap

aV aT Eventually, we want to discuss the Carnot cycle briefly, which will lead us to the notion of universal thermal efficiency. Historically this was essential for the understanding of the First and Second Fundamental Theorems of thermodynamics.

We use an arbitrary homogeneous substance and suppose that its state is determined exclusively by the two mechanical variables, the pressure p and the volume V, from which the temperature T can be computed by means of a general equation of state. A Carnot cycle is understood to be a closed path ry in the (V, p)-plane consisting of two isotherms and two adiabatics. More precisely:

(1) From A to B, the substance is expanded up to the volume VB through contact with a heat reservoir of temperature Ti while keeping the temperature fixed ("isothermally"). During this ?rocess it absorbs the quantity of heat Q1.

(2) The substance is taken out of the bath Tr at B and expanded to the point C without absorbing or emitting heat ("adiabatically"). The temperature drops to the value T2. (3) The material is put into a heat reservoir of temperature T2 and compressed isothermally from C to D, emitting the quantity of heat Q2.

8.2. Thermodynamical Systems in Equilibrium

291

(4) The material is taken out of the second heat reservoir and adiabatically compressed from D to A to reach the temperature T1 again.

It has to be stressed that we are supposing that the idealized processes discussed throughout this chapter are reversible. In the way explained here, the Carnot cycle describes a power engine (the temperature difference of the heat reservoirs is used to do work); reversing the process, it provides a thermal engine (work is done to further cool down the colder of the two heat reservoirs). By assumption, work is done exclusively in its mechanical form; the total work A is computed as

jdA

A :=

= -7

ipdV,

and, since -d(pdV) = dVAdp, this is equal to the area of the region bounded by the curve y in the (V,p)-diagram by Stokes' theorem. We integrate the First Fundamental Theorem

dE = dA + dQ over the closed path -y. The line integral of the exact form dE along 7 vanishes, since the inner energy again reaches the initial value at A:

0=

jdE ry

_ 5dA+ ¢ dQ. 'T

y

The thermal efficiency q is defined as the ratio of the work done and the added heat, q := A/Q1. But the line integral along dQ is precisely Q2 - Qi; hence we can write the thermal efficiency also as

q=1-Q1

8. Elements of Statistical Mechanics and Thermodynamics

292

As the Carnot cycle was supposed to be reversible, the entropy does not change, and the Second Fundamental Theorem implies the identity Q2

Qi

T2

T1

Jry T

The thermal efficiency only depends on the temperatures of both heat reservoirs.

n =

T1 - T2

T2

and not on the particular kind of substance or the particular construction of the power engine.

Exercises 1 (Gay-Lussac Experiment, 1807). Consider a fixed amount of an ideal gas which at time 1 has pressure pl and volume V1. The gas is heated while keeping the volume fixed until it reaches at time 2 the pressure p2. Then the gas is allowed to expand adiabatically up to the volume V3 ("overflow"). At the point 3. the gas reaches again the initial pressure pl. Prove that the entropy change between the states 2 and 3 is equal to

AS = S3-S2 = n-k-log

V2

independently of the way in which the physical process is realized in practice.

2. Derive from the Maxwell distribution the probability distribution for the speed lit' I I = (vi + v2 + V32)112 using spherical coordinates. 3. Consider two containers filled with the same ideal gas having equal temperature To and particle number N, but different volumes V1 and V2. The containers are then connected. a) Calculate the equilibrium temperature of the connected containers from the condition that the entropy remains constant during the connection process.

b) Explain why the work done is equal to the difference of the energies before and after the connection, and calculate the latter.

293

Exercises

4. Determine the quantity of heat Q absorbed by an ideal gas in the course of the Carnot cycle. Hint: One of the intermediate results is

Q = (T2-T1)(S2-S1) 5 (Van der W4aals Equation). The free energy of a real gas is heuristically defined as the sum of the free energy of the ideal gas F;d and an additional term depending on two constants a and b to be determined experimentally: V - nb n2a F = F;d - nkT log ( V F2

\

)

a) Explain why this Ansatz tends for large volumes to the free energy of an ideal gas, and why, on the other hand, it prevents the gas from being compressed unboundedly. b) Derive the equation of state (the so-called van der Waals equation)

(p +

n2a V

(V - nb) = nkT

as well as the formulas for entropy and energy. How can the result for ) the energy be interpreted physically? 6. The Hamilton function of an ideal gas in the ultra-relativistiiyc limit is n

(c ist the velocity of light). Calculate the free energy F, the entropy S, and the energy E.

7 (Planck's Radiation Law). An ideal photon gas in thermal equilibrium emits black-body radiation, which is described by Planck's law. We describe

each photon by a quantum mechanical oscillator and assume as a known fact that the energy levels of an oscillator with basic frequency w are

En = hw(n + 1/2). n E N . For photons, it may be supposed that these are distributed within each energy level (up to irrelevant factors of proportionality) according to the Gibbs distribution Qn = e-E^1kT

a) Calculate the mean energy,

x

x On - En

n=1

and conclude that the number n = [eF the mean occupation number.

On

n=1

1kT

- 11-1 can be interpreted as

294

8. Elements of Statistical Mechanics and Thermodynamics

b) The frequency w is related to the momentum of the photons by E = clIpIl = hw. Quantum statistics postulates that in the Maxwell distribution j 4p, the Gibbs factor e-`tIPII/kT is to be replaced by the mean QA[ of occupation number n. Conclude that the probability measure po I IpI I, up to some factors, is uPQaf

w2

- et-1k]

the spectral energy density c is defined as energy per volume unit, hence it is obtained after again multiplying by hw: hw 3

e=

efiW/kT _ 1

This is Planck's radiation law (Planck, 1900). c) Discuss the behavior of the curve c depending on the variable w and the parameter T; describe, in particular, how the position of the maximum changes with T. d) Discuss the limiting cases hw << kT (Rayleigh-Jeans law) and hw >> kT (Wien's law).

Chapter 9

Elements of Electrodynamics

9.1. The Maxwell Equations The Maxwell equations describe the impact of an electromagnetic field on a distribution of electrical charges in space as well as the interaction between the electric field E and the magnetic field B. These vector fields are timedependent vector fields defined on a domain 11 C R3,

E: 1lxR---+R3,

B: IlxR-.R3,

and the electric charge is described by a time-dependent density function

e: QxR -*R. The electromagnetic field induces a current of the electric charges, and hence also a time-dependent current density vector

J:S2xR-bR3. The operations of divergence and curl are supposed to be applied to the

spatial coordinates of the time-dependent vector fields.

First Group of the Maxwell Equations:

-

curl(E)

,

div(B) = 0.

Second Group of the Maxwell- Equations: 4c curl(B) = c + J, div(E) = 47rg.

-

295

9. Elements of Electrodynamics

296

Here c denotes the constant velocity of light. Since div(curl(B)) = 0 (see §2.3, Theorem 5), the continuity equation = 0

div(J) +

is an immediate consequence of the second group of the Maxwell equations. Now we want to write the Maxwell equations involving differential forms on

R3. For this, we use the transition from a vector field V to the associated 1-form wv discussed in Chapters 2 and 3. On a manifold Mm, wv is the 1-form defined by the equation (see §§2.3 and 3.4)

*wv = V J dMm, and the gradient and divergence satisfy the identities Wgrad(f) = df,

d(*wv) = div(V) - dAIm.

In JR3, there is an additional relation involving the curl of a vector field V (§3.11):

wcurl(V) = *dwv.

Passing now from the time-dependent vector fields E and B to the corresponding time-dependent 1-forms WE and WE, we can restate the Maxwell equations in the following equivalent form:

First Group of the Maxwell Equations: *dwE _

1 (9 .

c 8t

d(*wB) = 0.

(we).

Second Group of the Maxwell Equations:

le

47r

*dWB = - Cat (WE) + - wJ,

d(*wE) = 41rgo - dR3.

In these equations, the exterior derivative exclusively refers to the spatial coordinates, and not to the time coordinate. By applying Poincare's lemma, we can represent the closed 2-form *wB locally as the differential of a time-dependent 1-form A,

*wB = dA. Definition 1. The (locally) defined time-dependent 1-form A is called the magnetic potential.

The first Maxwell equation can be rewritten equivalently as d(wE + c

(A))

= 0.

9.1. The Maxwell Equations

297

Again, Poincare's lemma shows the local existence of a time-dependent realvalued function 0 with

A -do.

WE

c at Definition 2. The (locally) defined time-dependent function ¢ is called the electric potential.

The essence of the electromagnetic field is not the same in classical electrodynamics and in quantum mechanics. In classical field theory, only the measurable field strengths E and B are physically relevant; the potentials A and ¢ are auxiliary mathematical functions which turn out to be useful. In quantum mechanics, however, the electric and the magnetic potential acquire a physical meaning on their own; we illustrate this briefly by discussing the Aharonov-Bohm effect.

Remark (Aharonov-Bohm Effect). An infinitely extended coil S. through which a current flows, induces a magnetic field which is almost homogeneous

in its interior and vanishes outside of the coil. Consider a closed curve -y around the coil and denote by 0 the surface bounded by ry. Imagine that an electron is prevented from moving inside the coil by an infinitely high potential well. Then the remaining region ci - S of the plane is, from the point of view of the electron, no longer simply connected. The vanishing of the magnetic field B in this region implies that the 1-form A of the magnetic potential is closed in 11 - S, 0 = *wB = dA.

but nevertheless the line integral is not path-independent, since, for example, the curve segments ryl and rye of ry cannot be deformed into one another by a homotopy (see Theorem 9, §2.5). Instead, by Stokes' theorem, the integral

1A= JA

=

fdA

=

f

is equal to the magnetic flux through the entire region Q. Quantum mechanics describes the states of the electron by solutions of the Schrodinger equation called wave functions. Let V51 and v/2 be the wave functions of the electron along -ti and rye, respectively, while the current is switched off (i.e..

B = 0 in S). The covariance principle for the Schrodinger equation with respect to gauge transformations implies that, if the current is switched on, the wave functions have to become

I AI ,

01 exp L

7i

V2 = 02 exp [

,

I AJ

rr

9. Elements of Electrodynamics

298

We put a screen at Si and form the linear superposition of the two wave functions: B

+Gi + 1Pz =

'Pt exp [Z

2 ] + 02) eXp

A]

As we vary the enclosed magnetic flux 4'O, the absolute value IV)*, + t21 changes. This corresponds to a shift in the interference picture on the screen, and provides experimental evidence for the intrinsic meaning of the potential A.

A theoretically rigorous explanation of this effect (A does not exist globally on the region fZ!) can only be obtained within the theory of connections on principal S1-fiber bundles. The integral of A over a closed curve can then be defined correctly and measures the holonomy of the connection'.

If we put the potentials A and 0 at the beginning of our study, the electric and magnetic fields are determined by the equations WE

10 c a (A) -do, wB = *dA,

and the first group of the Maxwell equations is automatically satisfied. In these formulas, we view the quantities as time-dependent functions or forms in 3-space, and the Hodge operator * and the exterior derivative d refer to that space. We use the adjoint operator 5 of the exterior derivative d and note that in R3 the following formulas hold:

8 = -*d*

for 1-forms

and 8 = *d *

for 2-forms .

Then we can write the second group of the Maxwell equations as 8dA

1 8 2 (A) - 1 8 (do) + c2 8t2

c cat

w.J,

c

a 8A) c tit ( 1

O = 1aB

The Laplacian A on functions will be used in this chapter analogously to Chapter 2. Hence we have Ao = *d * do = -8d¢. 1As a physics reference we recommend: Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959),

485-491, as well as the more detailed discussion of this effect in the textbook by F Schwabl, Quantum Mechanics, Springer, 2nd rev. ed. 1995.

9.2. The Static Electromagnetic Field

299

9.2. The Static Electromagnetic Field If the electric and magnetic fields are time-independent, the Maxwell equations simplify: *dWE

*dwB

= 0,

d(*WB) = 0,

4a

d(*WE) = 4ap d1R3 .

- WJ, C

Hence the charge density P and the current density vector J do not depend on time, and J is a divergence-free vector field. This system of equations decomposes into a pair of partial differential equations for the electric field and another similar pair of equations for the magnetic field. We begin by solving the equations for the electric field. Using the locally defined electric potential WE = -do, this system reduces to the inhomogeneous Laplace equation

-AO = 47re . From now on we suppose that the support of the charge density is compact. Then it is to be expected that the electric field E(x) generated by this charge distribution tends to zero at infinity. Under this condition, the electric field is uniquely determined.

Theorem 1. Let the support of the charge density p be compact. Then there exists precisely one electric field E such that

div(E) = 4ae, curl(E) = 0 and IjE(x)II - 0 for Jlxii with the electric potential

oc. This electric field is given by E = -grad(4) 0(y)

OW

dy,

11x - Y11 R3

which is also called the Newton potential for the density e.

Proof. Consider two electric fields El = -grad(01), E2 = -grad(O) generated by one and the same charge density p. The difference u := 01 - 02 of the electric potentials is a harmonic function with a gradient vanishing at infinity,

Au = 0, 1Igrad(u)(x)iI - 0 for IIxil -+ oo. Since i (Ou/Ox') = 8(Du)/ax' = 0, the partial derivatives au/8x(1< i < 3) are harmonic functions on 1[t3 also vanishing at infinity. Liouville's theorem for harmonic functions (see Theorem 42, §3.10) immediately implies that all partial derivatives vanish identically. Hence u is a constant function,

and the electric fields El =- E2 coincide. There exists at most one electric

9. Elements of Electrodynamics

300

field with the required properties, and it remains to be proved that the electric field E = -grad(O) with potential

OW = -

f

P(y)

lix - yII

dy

R3

is a solution for the problem. To see this, we have to show that O(x) is a smooth function in R3, that it solves the inhomogeneous Laplace equation

-0O = 41rp, and that the gradient l grad(O)II -' 0 at infinity in R3. We introduce spherical coordinates around a fixed point x = (x1, x2, x3) E R3 (0 < cp < 27r, -a/2 < ip < 7r/2): y2 = x2 + r sin c p cost

y1 = x 1 + r cos cp cos ? / i ,

,

y3 = x3 + r sin Eli .

The volume form of euclidean space is described in these coordinates by the following formulas:

dR3 = dy1 A dy2 A dy3 = r2 cos /i dr A dcp A d/i. For every positive number e > 0, the function 1

Ay) :=

yii3_e

IIx is integrable in a neighborhood of its singular point. Integration over the ball D3(x,1) with center x E R3 and radius 1 yields 1

r

J

2,r K/2

ffJ r3

-E cos i,i dr dcp dzb = 4a/e.

f (y) dy =

0

D3(x,1)

0 -7r/2

Because of the compactness of the support of the charge density g(y), the function c(x) and its first and second partial derivatives are smooth, x' y` ao e /' dy = - /' P(y) ayi (ox P(y) (x) -

ax' -

J

IIxyII3

YII) dy

= f ay= (y) 1 dy II x yII R3

Differentiating once again leads to the formula

- AO(x) = lim lim

1

f 0(P)(y) '

MM-00 E-0 D3(x.M)\D3(x,e)

IIx

dy. yII

Applying Green's second formula for sufficiently large radius M, we can rewrite the integral over the spherical shell D3(x, M)\D3(x, E):

f

o(P)(y)

D3(x,M)\D3(x,e)

FIX

1 yII

f P(y) ' b IIx 1 y1I

D3(x,M)\D3(x,M)

9.2. The Static Electromagnetic Field

- f [(y)(gy1 1

Y11,

N(y))

301

-

8D3(x,e)

IIx

\grady(y),

1 YII

N(y))] dy.

The function 1/IIx - yUI is harmonic in the spherical shell: hence the first integral vanishes. Moreover, the normal vector field of the sphere 8D3(x, e) of radius a is given by the formula N(y) = (y - x)/E, which immediately implies 1

(grads

1

IIx - Y1I ,

V(y))

E2

Inserting these expressions, we obtain

f

OP(y) II

D3(x,A1)\D3(x.e)

x-

1 y1I

dy =

J (grad e(y), y-r)dy . f y(y)dy+ 2 e

2

aD3(x,e)

aD3(Xx)

The second integral on the right-hand side is bounded by (1 /E2) - e - max{ Ilgrad oII } vol(8D3(x, E)) = 4;r - e max{ Ilgrad oII } ,

and hence it does not contribute in the limit as I

0. Together, this yields

-O0(x) = limo F J B(y)dy = 4irp(x), 0D3(x.e)

where we used the mean value theorem of integral calculus in the last step.

We estimate the length of the gradient of the electric potential using the formula for the partial derivatives 8o/0x` and for points lying far out. To do so we choose a radius R > 0 such that the ball with center 0 E R3 contains the support of the charge density. If now IIxII > R, then 2 3

Ilgrad6(x)II2 = i=1

f 0(y)Ily-x113dy x{ _ yi

D3(0,R)

<

1

3-max{p2(y):yER3}.4rr

3

dist2(x. D3(0; R))

We conclude that the length of the gradient IIgrad 4(x)II decreases like 1/IIxII for IIxII

oo

Remark 1. Obviously, the preceding theorem and its proof remain valid for a charge density p which no longer has compact support, but, together with its derivatives, vanishes sufficiently fast at infinity.

Example 1. Consider for a fixed point xo E R3 a charge density e which is concentrated in a small ball around it,

ee(y) _

fe

(vol(D3(xo,E)))-1 0

if y E D3(xo,E), otherwise.

9. Elements of Electrodynamics

302

We calculate the electric potential: Lot(Y)

(x)

-e

dy =

I

vol(D3(xo, e)

IIx - Y11

dy, IIx - Y11

D3(xo,e)

R3

and in the limit as e -+ 0 we obtain lim ,

E-o `

(x) -

-e IIx - x011

In this way, the Coulomb potential arises naturally as the field generated by an electric point charge. Let, similarly, n points xi, ... , x,, with charges e1, ... , e be given. Then they generate the potential n

O(x) = - E IIx ej- xcll i=i

Now we turn to the solution of the two partial differential equations for the static magnetic field B(x) and consider again only the case that the divergence-free current density vector has compact support (see the remark above).

Theorem 2. Let J be a divergence free vector field with compact support in R3. Then there exists precisely one magnetic field B such that 4a

div(B) = 0, curl(B) =

c

J

and IIB(x)II - 0 for IIxii - oo. This magnetic field is given by B = curl(A) with the magnetic potential A(x)

l cJ R3

dy. IIx -Y11

Proof. Let B1 and B2 be two magnetic fields with the stated properties. Their difference V := BI - B2 is a divergence- and curl-free vector field,

div(V) = 0, curl(V) = 0, whose length II V(x) 11 tends to zero for IIx11 -+ oo. By Poincare's lemma, such

a vector field can be represented as the gradient of a harmonic function f ,

V = grad(f) and Of = 0. The partial derivatives Of /8x' (1 < i < 3) are thus bounded harmonic functions on the entire space R3, and by Liouville's theorem they are constant. This immediately implies V = Bl - B2 = 0, and this observation shows that there can be at most one magnetic field with the required properties. By

9.2. The Static Electromagnetic Field

303

the same arguments as in the proof of the preceding theorem, the magnetic potential AY) dy c A(x)

J

R3

11X - Y11

is smooth. It remains to be shown that it is divergence-free. The partial derivatives of the components A = (Al, A2, A3) of the magnetic potential are P J`(y) (x' -1/') ON c R3

dy,,

llx-yll3

and hence

div(A) =

Z'

J

(J(y), IIx - y113

R3

)dy = I f (J(y), grad, (IIx R3

)dy.

1 yIl)

Using the formula

divy(lix

1 Y11 J(y)) = (J(y),g'ady(llx 1 y1I)) + I1x 1

YIIdiv(J)

and the fact that J was supposed to be divergence-free, we obtain

div(A) =

f divy(llx 1 YIIJ(v))dy = 0 c

R3

by Green's formula, since the vector field J(y) has compact support. The curl of the vector field B = curl(A) can now easily be calculated (see Exercise 6.b, Chapter 2):

curl(B) = curl(curl(A)) = grad(div(A)) - 0(A) = -0(A) Here the Laplacian is to be applied component-wise to the vector field A, and the proof of the preceding theorem yields the result

curl(B) = -i(A) =

47r

c

J.

Example 2. Let the current density vector J be constant in a domain ft C H3 and zero outside this domain. The magnetic potential and the magnetic field B = curl(A) are then determined by the formulas A(x)

c

(fn

II X

'YII) J' B(x)

c J " f II(x - 11 dy. n

The magnetic field induced by an electric charge distribution with constant current density vector is perpendicular to the direction of the current flow. This fact is called the Biot-Savart law.

9. Elements of Elect rodvnamics

304

Finally, we interpret both theorems as the Hodge decomposition of a 1-form on R3 (see Corollary 5, §3.11). Let wl be a 1-form on II23 with compact support or, more generally, with rapidly decreasing coefficients. Applying Theorem 1 to the function p = *d(*wl), we obtain a closed 1-form wE with

dwE = 0 and * d(*wE) _ *d(*w') . The form wE also tends to zero at infinity, and hence, by Theorem 2, the difference J = w' - WE can be represented as

*dw' = J = w'

- wE

with a 1-form wa vanishing at infinity. This observation leads to

Theorem 3 (Helmholtz's Theorem, Hodge's Theorem). Every smooth 1form w' in R3 with compact support, or, more generally, with rapidly decreasing coefficients, can uniquely be represented as the sum of a closed and a coclosed form.

w' = WI +*dwg,

L4 = 0.

wF and 4 both tend to zero at infinity.

Remark 2. Comparing this result with Corollary 5 in §3.11, we stress that, in I3. the harmonic part does not occur in this decomposition. The reason for this is that there are no harmonic 1-forms in 1R3 having compact support or tending to zero sufficiently fast at infinity. If w' were such a 1-form, then, by Poincare's lemma, we could represent w' as the derivative of a harmonic = 0. The partial derivatives Of 149x' would then be function, w' = df,zf, bounded harmonic functions, and thus constant. If then the length of w' tended to zero at infinity, then these partial derivatives would vanish and we could conclude that wl = 0.

9.3. Electromagnetic Waves An electromagnetic wave is an electromagnetic field in the vacuum (p = 0. .1 = 0). The Maxwell equations in this case read as

div(B) = 0, div(E) = 0,

curl(B) = 149E 8t

,

curl(E)

149E

c 8t

,

which implies that the components of the electric and magnetic fields have to be solutions of the wave equation

AB = 1 02B

c2 &2 '

AE _

1 492E

cz 8t2

.

305

9.3. Electromagnetic Waves

If the wave is known at a certain time t = 0 and we want to study its propagation in time, we have to solve the Cauchy initial value problem for the wave equation:

Du =

102 c2 at2 ,

u(x, 0) = uo(x), - (x, 0) = ui (x) .

For smooth initial conditions uo and ul, this problem has precisely one solution, which we are going to describe now. To do so we need the spherical

mean of a function V : 1R3 --i R. This is understood to be the function IV : 1R3 x 1R+ -+ 1R which computes the mean value of cp on an arbitrary 2-dimensional sphere S2(x; r) with center x E R3 and radius r > 0, (1V)(x,r)

4Trr2

f p(y)dy

S2(x,r)

Theorem 4 (Poisson Formula for the Wave Equation). For any two smooth initial conditions uo, ul : 1R3 - R, the Cauchy initial value problem for the wave equation 1

Du =

2

0)

0) = ui (x)

u(x, = uo(x), at has precisely one solution on the space 1R3 x 1R+. Moreover, this solution can be computed explicitely: c2 at2 '

u(x, t) =

(x,

(t (Iuo)(x, c t)) + t (Iul)(x, c t). ttII to+

A

(xo, to)

R3

r- (xo, to)

Remark 3. The value u(x, t) of the solution for the wave equation depends only on the behavior of the initial values on the boundary of the base of the backward light cone,

r (xo, to) := {(x, t) E 1R3 x 1R+ : Iix - roil < c It - toe, 0 < t < to} , which is a two-dimensional sphere. Therefore, every wave in 1R3 generated by initial conditions with compact support has a forward as well as a backward wave-front (Huygens' principle).

9. Elements of Electrodynamics

306

Proof of Uniqueness for the solution. We introduce the notation 1 012

A- cz for the wave operator and

Eu(x,t) := 1(a l+(cat J 3

z

19U

Jz

J)

for the energy. Then the following easy identity holds:

au

a

3

a

au au

E( u)

Now let u be a solution of the wave equation with initial conditions u(x, 0) _ 0 and au(x, 0)/at = 0, and denote by FT (xo, to) the truncated cone described

by the condition 0 < t < r. By integrating the stated identity, we obtain

a

rr

E(u) = 2 at

r 3 a au au J r,

axe

t axe

Gauss' theorem (Theorem 27, §3.8) allows us to transform the volume integrals into surface integrals: 3

f E(u) ' (N, atj = 2

at axj \N' axj

er, '=1

arr

The boundary 81'7 consists of three parts. On the first part (t = 0), the energy E(u) and the derivative au/at vanish; this term does not contribute to either side of the equation. On the second boundary piece (t = r), the normal vector N is parallel to a/at. Hence the integral on the right-hand side of the equation vanishes, whereas the integral on the left-hand side is non-negative. On the third part of the boundary, the lateral surface of the truncated cone, we calculate the scalar products

_

-

zo C ( a _ CN'ax) Ilx - xoll 1 + i+c \N' Thus we can transform the difference of the left-hand and the right-hand

a

xJ.

integrand into an expression which is non-negative, too: E

(

a

\N' at / -

2

au au

a

at axi \N' axi /

-

c

i+

F-

2

au x) - xo

On

cat l I x - xo l l

ax)

All in all, the integral formula implies that the energy E(u) vanishes on

the upper boundary (t = r) of the truncated cone E. The height r of the truncated cone can be varied, and thus E(u) vanishes identically in the

307

9.3. Electromagnetic Waves

interior of the cone r- (xo, to). There the function u(x, t) is constant, and, looking at the initial conditions, identically zero.

Proof of the existence of a solution. We calculate the derivatives of the spherical mean with respect to time:

(Iul)(x,ct) =

Jui(y)dy =

4nc2t2

f u1(x+ct z)dz,

41r

52

S2(x,Ct)

where S2(x, Ct) is the sphere with center x E R3 and radius ct. Denote by S2 the unit sphere. Then Green's formula implies

8(lul) at

r

s

(x, ct) = c J 4w

47rct

2

z

C?u l

axjj (x + ct. z)dz ax-

j=1

a"1(a)da 2 f N(a), axj ' axJ

=

l 2J

41rrt

Aul (y)dy

D3(x,ct)

S2(x.ct)

Introducing spherical coordinates (r, a) E (0, oo) x S2 in the three-dimensional ball D3(x, ct) and applying the formula d]R3 = r2 dr A da, we obtain ct

a(at

1)

(x, ct) =

jr2jAU,(X+ra)da-dr. 4x 2 0

S2

This leads to Ct

(t (Iul)(x,ct)) = (Iul)(x,ct) + 41rct f 0

r2

f Dul(x+ra)da dr, S2

and, after further differentiation, 2 2

JAu,(x+ct.a)da = c20(t' (Iul)(x, ct)) (t. (Iu1)(x, ct)) = 4t S2

Thus w(x, t) := t- (1ul) (x, ct) is a solution for the wave equation with initial conditions 2

w(x, 0) = 0 = 02 x, 0)

and

at(x,0) = (Iul)(x,0) = ul(x).

The derivative Ow/8t is also a solution of the wave equation, and it provides the second summand in the asserted formula.

We have thus solved the wave equation for a scalar function on R3. The expression for the vector valued functions E and B follows from a tedious but routine computation. The explicit formula may be found in Exercise 3. Now we turn to the inhomogeneous wave equation u = f with a smooth

9. Elements of Electrodynamics

308

function f : JR3 x R+ -s R, and will solve this for the initial condition ud = 0 = ul. Combining the resulting solution with a solution for the homogeneous wave equation with arbitrary initial condition constructed in the preceding theorem, we will arrive at the solution for the inhomogeneous wave equation Du = f with arbitrary initial condition u0, u1. The retarded potential of a smooth function f : R3 x lR --> JR is defined by the integral

Rf(x,t) := -

f(y,t-r/c) dy

I

r

41r 11x-Y11
with the distance function r:= uix - y1l.

Theorem 5. Let f : R3 x R+ - R be a smooth function. Then the retarded potential R f is a solution of the Cauchy initial value problem for the inhomogeneous wave equation, Du = f,

u(x,0) = 0 = 8 (x,0).

Proof. Passing to spherical coordinates with center x E R3, we can represent the retarded potential as

Rf(x,t) _

fJ s2

Introducing the parameter r := t - 1: and the function

p(x,t,r) :=

t47rJ 32

we obtain from p(x, t, t) = 0 the following formulas for the retarded potential and its derivatives: tr

tr

Rf(x,t) =-c2J p(x,t,r)dr, 8t(Rf)(x,t) = -c2J 0

(x,t.r)dr,

0

2

5jZ(Rf)(x,t) = -c2 J ate(x,t,r)dr-c2L(x,t,t). 0

Calculating the derivative of p with respect to t yields

5'(x, t, t) = Irf f (x, t) da = f (x, t) S2

,

9.3. Electromagnetic Waves

309

and, summarizing, we obtain t

a

(x,t,T)dT-c2 f(x,t)

22(Rf)(x,t) = -c2

.

0

Applying the wave operator to the retarded potential leads to the formula t

D(Rf)(x,t) = f(x,t)-c2 f Dp(x,t,T)dr. 0

If the parameter T is fixed, the function p(x, t, T0) =

t 4i 0 f f (x + c(t - TO)a, T0) d S2

is a solution of the homogeneous wave equation. This immediately follows from the preceding theorem, since p(x, t, TO) is defined as a spherical mean. Hence the retarded potential solves the inhomogeneous wave equation. 0

The inhomogeneous wave equation Du = f may be interpreted as follows. For simplicity, suppose that f describes the current generated by a charged particle moving along the world line y. At the point A, the particle produces an electromagnetic field which moves at the speed of light along the surface of the light cone with vertex A. Hence the field in B depends on the motion of the particle in A. In this way, the principle of causality appears as a mathematical consequence of the inhomogeneous wave equation, and its retarded solutions were first described by the physicists Lienard (1898) and Wiechert (1900).

x

9. Elements of Electrodynamics

310

The solution of the two-dimensional wave equation can be constructed starting from that of the three-dimensional wave equation. To this end we transform the spherical mean of a function ¢(x', x2) depending only on the first two spatial variables into a two-dimensional integral by parametrizing the sphere S2 (x, ct) C R3 through

yl = xl + a1, Y2 = x2 + a2, y3 = x3 +

c2t2 - (al )2 - (a2)2,

where (al)2 + (a2)2 < c2t2. A straightforward calculation shows that the volume form of the sphere is described by the formula dS2

=

t2 _

(1) - (a2)2 dal n dal,

from which we obtain the following expression for the spherical mean:

I (xl x2 t) =

1

29rct

1J

_(x1 - a1, x2 - a2) dal n dal. c2t2 - (al)2 - (a2)2

D2(O,d)

We treat the retarded potential of a function only depending on the first two spatial variables and time analogously f (x1, x2, t):

Rf (xl, x2, t)

_

f xl - al, x2 - a2, r

C

21r

J

(t - r)2 - (a1)2 - (a2)2

dal nda2ndr

.

Inserting the resulting expressions into the formulas of the preceding theorem, we obtain a representation of the solution of the inhomogeneous twodimensional wave equation with prescribed initial conditions uo(xl,x2) and

u1(x',x2) as an integral over the interior of a disk in R2. From this we conclude that a wave in R2 generated by initial conditions with compact support has a forward front, but it does not have a backward front. This effect is well-known, for example in the case of water waves. Stated differently, this means that Huygens' principle (see Remark 3) does not hold in two dimensions (in fact, it can be shown to fail in all even dimensions). Now we turn to the question of which conditions rotating an electromagnetic

wave lead to another electromagnetic wave. Let an electromagnetic wave E(x, t), B(x, t) be given. We rotate these vector fields in the 2-plane spanned by the two vectors through the angle 1/i(x, t):

E` (x, t) =

cos 1Ji(x, t) E(x, t) + sin '(x, t) B(x, t)

,

B'(x, t) = - sin 1'(x, t) E(x, t) + cos?i(x, t) B(x, t)

.

We discuss the conditions to be satisfied so that the pair (E', B') becomes an electromagnetic wave. Necessarily, ip has to be a solution of the eikonal

9.4. The Relativistic Formulation of the Maxwell Equations

311

equation

8t-) - Ilg'ad('+6)fl2 = 0. (2

1

c2

Theorem 6. (E*, B') is an electromagnetic wave if and only if the phase function 0 satisfies the following conditions:

(1) (grad(tG), B) = 0 = (grad(tG), E);

(2) - 00 ' B = -grad(O) x E; (3)

c

E = grad(O) x B.

Proof. The vector fields E and B are divergence-free, and hence the divergences of E' and B` are easily calculated:

div(E*) = - sin i' (grad(,O), E) +

dcos'. (grad('+G), B) ,

div(B') = - cos lO (grad(t/i), E) - sin O (grad(O), B) . Thus the vanishing of these divergences div(E') = div(B') = 0 is equivalent to the condition (grad(t1), E) = 0 = (grad(Vi), B). Making use of the formula

curl(f V) = grad(f) x V + f - curl(V), which generally holds in R3, we write down the Maxwell equations for the pair (E', B'). This leads to the following pair of equations: sin ,p

grad(O) x E - c

sin t/' (_grad(1iL) x B +

-6T

-C -N

B l = cos 10 _ -

E - grad(O) x B)

5T

. E) = cos 7[i (

B + grad(t/') x E I

which are equivalent to the second and third conditions of the theorem. 0

9.4. The Relativistic Formulation of the Maxwell Equations Consider Minkowski space R3.1 with coordinates (x, y, z, t) as well as the pseudo-Riemannian metric of index 1 determined by g :_ (dx1)2 + (dx2)2 + (dx3)2 - c2dt2,

the so-called Minkowski metric (see Example 35, section 3.11). The 1-forms j1 , dx2, dz3 and c dt form an orthonormal frame. Computing the Hodge

9. Elements of Electrodynamics

312

operator * of 183.1 on 2-forms leads to the following result:

_ -dx3 A c dt, *(dxl A c dt) = dx2 A dx3 ,

*(dx' A dx2)

= dx2 A c dt, = -dx1 A c dt,

*(dx1 A dx3)

*(dx2 A dx3)

*(dx3 A c dt) = dx' A dx2 .

*(dx2 A c dt) = -dx1 A dx3 ,

If w is a 1-form on 1R3, the Hodge operators * of euclidean space R3 and * of Minkowski space 183.1, respectively, are related by *w.

In R33 we define the 2-form called field strength form by

F := and the 1-form called density form by 1

J :=

C

wr-

Theorem 7. The 2-form F has the following properties: (1) IIFI12 =

IIBI12-IIE112.

(2) (F,*F) = 2(E, B). (3) The first group of the Maxwell equations is equivalent to

dF = 0. (4) The second group of the Maxwell equations is equivalent to

*d*F = Proof. We calculate the exterior derivative dF of F in 4-space:

dF = (*&B)+a( =

(d(wE)+

B)

Adt

18(*OtwB))

and see that dF = 0 is equivalent to the pair of equations dR3 (*wB)

= 0 and e (WE)

8

(*wB)

5i

Similarly, we obtain

d*F=d(*w E - WB A cdt) =

C

dR3 (*wE) +

(*wE) A dt - 0 (wB) A cdt,

and, after a slight reformulation,

*d*F = *dR3

18

(wB)-c8t(WE)

9.4. The Relativistic Formulation of the Maxwell Equations

313

Summarizing, * d * F = 4a J is equivalent to d(*WE) = 47rp and

18

* d(wB) =

4c

c 8t (wE) +

wj.

Remark 4. The 2-form F is a closed form in ]R3.1. Hence by Poincare's lemma there exists a 1-form A such that

dA = -eF. c The 1-form A comprises the electric as well as the magnetic potential:

F=

c 5i dR3.'(A

= and thus A = -(e/c)A + e 0 dt.

The isometry group of the Minkowski metric g is 0(3, 1), the Lorentz group. It consists of all linear transformations L :183,1 -+ R3,1 which leave g invariant. If we represent g by the matrix

A= -C2

0

then 0(3, 1) can be identified with the (4 x 4) matrices L satisfying

A. Its Lie algebra o(3, 1) is spanned by the matrices X with

Lt = 0. Of course, 0(3, 1) contains the 3-dimensional group of rotations of euclidean space R3 as a subgroup, embedded as the upper left 3 x 3 matrices

0

0

M E 0(3,R)

1

1 and generated by the skew-symmetic matrices L1, L2, L3 described in §6.3, Example 14. The missing part of the Lie algebra o(3, 1) is spanned by the three elements { I.

1

0

B'

0

'

0

B2 :=

000

L

0

f

B3 :=

0

0

00J'

0

0

00G

0

with commutator relations [B1, B2]= c2L3,

[B1, B3[ _ -c2L2,

[B2, B3[ = c2Lg,

9. Elements of Electrodynamics

314

[L1, B2] _ -B3,

[L1, B3] = B2,

[L3, B1] _ -B2,

[L2, B1] = B3,

[L3, B2] = B1,

[L2, B3] _ -B1,

[Li, B,] = 0.

In the same way as L 1, L2, L3 generate ordinary rotations, the elements B1, B2, B3 induce hyperbolic rotations, for example,

e OBI

-

cosh(c8)

0 0 sinh(cO)/c

0

1

0

0

0

1

c sinh(c 8)

00

0 0

cosh(c 8)

The defining condition for L E 0(3, 1) implies that L has determinant 1 or -1. The hyperbolic rotation e8Bi and the time reflection diag(1,1,1, -1) realize these values of the determinant. Hence 0(3, 1) has two connectivity components.

Theorem 8. (1) The inhomogeneous Maxwell equations dF = 0 and bF = 41rJ are invariant under isometries of the Minskowski metric, the Lorentz transformations. (2) The homogeneous Maxwell equations dF = 0 and d * F = 0 are invariant under conformal changes of the metric. (3) The quantities I IFI I2 and (F, *F) are invariant under Lorentz transformations.

Proof. An element eox E 0(3,1) acts on differential forms and vectors by its differential X E o(3, 1). The 2-form F transforms into its pullback X*(F). But since pulling back commutes with exterior differentiation, dF =

0 is equivalent to d(X*F) = 0, which establishes the Lorentz invariance of the first Maxwell equation. For the second equation, observe that the volume form dM of any pseudo-Riemannian manifold M is invariant under

isometries of the metric; hence the adjoint of d, the coderivative 0, also commutes with pullbacks by isometries. Thus, OF = 47rJ is equivalent to X*(bF) = 47rX*(J) and b(X*F) = 4irX*(J). This shows the Lorentz invariance of the second Maxwell equation. Now consider a conformal change of the metric on a pseudo-R.iemannian

manifold M, y = e2f g for some function f. The exterior derivative does not depend on the metric; the first Maxwell equation is thus trivially conformally equivalent. An orthonormal basis el,... , e with dual basis of 1-forms

Qn transforms into

e-fel,...,e-fe, and efv1,...,efa,,. If M has even dimension n = 2k, we have the remarkable effect that the Hodge operators * and * with respect to the metrics g and g coincide on

9.4. The Relativistic Formulation of the Maxwell Equations

315

differential forms in half the dimension of M. For example,

*(a,A...Aak) = -'ak+1A...Aa2k, *(efaiA...Aefak) = E'efak+1A...Aefa2k

for some irrelevant sign e = f1, so that k factors of of can be cancelled in the action of *, which implies the claim. Its follows for the 2-form F on 4-dimensional Minkowski space that d * F = 0 is conformally invariant. 0 The conformal group can be identified with O(4, 2), whis has dimension 15. The fact that only the homogeneous Maxwell equations are conformally invariant was the main reason why Lorentz invariance has been established as the fundamental invariance property of electrodynamics. However, the full conformal invariance is useful when dealing with homogeneous situations. The invariance of the scalar quantities I IF112 and (F, *F) is of practical importance, too; it means. for example, that E and B stay orthogonal if they were orthogonal in some frame of reference. As a 2-form, F is an element of 112(R3"). There exists a purely algebraic,

Lorentz invariant map from 112(R3.') into the space of symmetric (2,0)tensors. Apart from normalizations and an additional multiple of the metric, the image of F under this map is known as the Maxwell stress tensor, which we will now discuss. So let F be any 2-form on Minkowski space, and e1, ... , e4 an orthonormal basis, i.e., satisfying g(e;, ej) = ej d;j with 1 = el = s2 = E3 = -64. We define

TF := where O denotes the symmetric product of 1-forms (see 5.3). This is an SO(3,1)-invariant, symmetric (2, 0)-tensor whose trace is a multiple of IIF112.

In the example at hand, we may choose

el =

axl,

e2 = axe' e3 =

ax3

and e4 =

.

The 2-form F is given by F = WE A c dt + Bl dx2 A dx3 - B2 dx' A dx3 + B3 dx' A dx2 .

Hence one computes (x4 := cdt)

TF = (Eicdt - B2dx3 + B3dx2)2 + (E2cdt + B1dx3 - B3dx1)2 + (E3cdt + Bldx2 - B2dx1)2 - (Eldxl + E2dx2 + E3dxa)2 4

_:

2 dx`OdxJ.

9. Elements of Electrodynamics

316

In order to obtain a trace-free endomorphism, one typically defines as the Maxwell stress tensor the combination

Tit :_ -2TF + (F, F) (dx' C dx' + dx2 ^ dx2 + dx3 ®dx3 - c2dt O dt). Its importance stems from the fact that its components have a physical interpretation. For example, its pure time component T4'4'

= -(11E112 + 1IBI12)c2dt2

is the energy density of the electromagnetic field, and the mixed contribu-

tions (i = 1.....3)

T"' = 4(ExB)idx'0,cdt are the components of the Poynting vector E x B. which has the dimension of an energy flux density. Since Ta' contains basically the same information as F, it is plausible that one can rewrite the Maxwell equations in terms of TMM solely. For their derivation, we refer to classical textbooks in eletrodynamics. The electromagnetic potential A is a real-valued 1-form defined on Minkowski space. We now view the real numbers as the Lie algebra of the compact onedimensional group U(1). in which the only quantity describing the electron which is relevant in Maxwell's theory is encoded---its electric charge. Passing now to the description of more complex elementary particles which carry

not just electric charge, but in addition other characteristics (color, ... ), we have to replace the one-dimensional group U(1) by a higher-dimensional compact Lie group G. If we also want to include gravitation in the model, as understood in the sense of general relativity, flat Minkowski space has to be replaced by a pseudo-Riemannian manifold. Combining all this, we are led to define a "generalized electromagnetic field with gravitation" as a 4-tuple (Al, g, G, A) consisting of a pseudo-Riemannian manifold, a Lie

group G, and a 1-form A : T(M) -+ g with values in the Lie algebra g of G. The non-commutativity of the group G will become essential in the sequel. The field strength FA is described by a g-valued 2-form, and to the Maxwell equation in the vacuum there corresponds the so-called Yang-Mills equation,

FA := dA + 2

[A, A],

DA(* FA) := d * FA + [A, *FA] = 0 .

The possibly complicated topological structure of the base space Al may lead to a situation in which the generalized electromagnetic potential A is not a globally defined 1-form on M. This leads to topics studied in the differential geometry of principal fiber bundles and the theory of connections.

9.5. The Lorentz Force

317

With their help, the models discussed in the physics of elementary particles can be formulated and studied rigorously.2

9.5. The Lorentz Force In this section, we discuss the equation of motion for the Lagrangian of a massive particle in an electromagnetic field as introduced in Example 11 of Chapter 7. We show that it is a natural generalization of the Lorentz equation, and discuss its solution for a few electromagnetic fields, i.e., a time-independent homogeneous field and a Dirac monopole. Frenet's formulation of curve theory will prove to be a suitable tool for the description of the particle's motion. Let (Mm, g) be a pseudo-Riemannian manifold, A a 1-form on it, and consider the Lagrange function

L(v) =

2g(v, v) - A(v) .

Theorem 9 (Generalized Lorentz equation). A curve e(s) : R - Mm is a motion of the Lagrange system (Mm, L) if and only if it satisfies

mds = V(eJdA), where the right hand side is meant to denote the vector field associated with the 1 form e J dA relative to the Minkowski metric.

Proof. We use local coordinates x1, ... , xm on Mm. The Lagrangian can then be rewritten as m

L(xl,...,xm,2....... m)

2 i,j=1

i=1

By definition, e(s) : ]R -p All is a motion of the Lagrange system (Mm, L) if and only if it fulfills for all k between 1 and m the Euler-Lagrange equation ds

(8i (e(9))) = a k (O(s)).

Since 6(s) = (dx1 /ds, ... , dxm/ds), the left and right sides can be computed to be OL

m

No j dxi dxj

8xk W M =

2

8xk ds ds

to

8A; dxi

8xk ds t

2See Thomas Friedrich, Dirac Operators in Riemannian Geometry, Grad. Studies in Math. vol. 25. AMS, Providence, 2000.

9. Elements of Electrodynamics

318

and d aL agile dxi dx' d2xi (aik (9(s)) I = m>2 ds axle ds ds + m>9ik ds2

aAk dxi i

axi ds

Equating them and rearranging some terms, we obtain the equation (*)

[aAk

aAi dx

axi - axk ]

ds

e m

9ikd2xi

ds2

r99ik

+m

i

ax; i,j

1 agi; dx' dx' 2 axk I ds ds

Hence it remains to interpret both sides geometrically, as done in the formulation of the theorem. We begin by recalling the definition of the covariant derivative as discussed in §5.7, Definition 21. From there, we know that the covariant derivative may be expressed in terms of the Christoffel symbols associated to the chosen local coordinates as M ye

_

d2xi a

ds = m E d82 axi

+m

dxi dx' a ds ds axi

But the Christoffel symbols can by computed from the metric (§5.7, Theorem 42 ff.): k

r+

-

1 [: km agim a97m 89ii 9 8xk + Oxi - Oxm m

2

hence one checks that the right-hand side of the equation of motion (*) is equal to

r ve a l

g 'n ds ' axle Similarly,

dA =

ij

ez dxi A dxx

implies that dA (e,

8

aAk

axk)

axi

i

aAi dxi axk,

d

which is just the left-hand side of (*). We conclude that equation equivalent to g

8 a md,8xk) = dA (k, axk

Since this has to hold for all k, the/claim is proved.

1

0

9.5. The Lorentz Force

319

The most important situation where this result can be applied is the case that Mm is the flat Minkowski space R3.1 with its pseudo-Riemannian metric. As 1-form on R3'1, we choose the form A introduced in the previous section, which is composed of the magnetic 1-form A on the euclidean space R3 and the electric potential ¢:

A = -eA+ecdt = -e(A1dx1+A2dx2+A3dx3)+eodt, and it satisfies dA = (-e/c) F. We emphasize that time t is viewed as the fourth coordinate of space, whereas the curve parameter s of p is interpreted as the proper time of the particle. The curve p(s) = (x1(8), x2(s), x3(s), t(s)) is said to be parametrized in proper time if its tangent vector is normalized

to length -c2,

2: = d-

c2

11

In classical mechanics as discussed in Chapter 7 (the Newtonian limit), all velocities were so small compared to the velocity of light c that it was safe to identify s with time. We shall henceforth abstain from denoting derivatives with dots or primes, in order to avoid confusion. By velocities, we mean the time derivatives vi := dx`/dt with euclidean length v2 := vi + v2 + v32. The parameters s and t are related through the relativistic -y-factor. To compute it, we express the length of do(s)/ds as

3 [i)2

2

d

oII

=

` ds

-c2

[]2 = ds

3

dxt dt 2-c2 ds [ dt ds]

[dt]2

dt 2 = (v2 -c2) [ds]

and deduce from its length normalization that dt 1 _ ds . 7, = 1/ry . and ds = dt 1 - v2/c2 Thus, it is comfortable to rewrite do/ds as do

ds -

al

r a a a +v3a7X3 + at Y v1ax1 +V2 57X2

The right-hand side of the generalized Lorentz equation now reads, using the definition of F, c TS

WS J

j F = +-_r [_(v. E)cdt +

(cE1 + (v x B)i)dx`

where v E denotes the euclidean scalar product and v x B the vector product in R3. Its dual vector is

V(doJdA) = ry ds

c2

a +(eE;+e(vxB)i) c at ax'

320

9. Elements of Electrodynamics

Since the left-hand side of the generalized Lorentz equation is

ddg _

ddp

md-ds = mrydt ds we conclude that its spacelike components yield after simplification the classical Lorentz equation (1)

mdt(ryv)

= e E+ _v x B,

and its time component leads to mdry/dt = +(e/c2)v E. This equation describes in fact the change in time of the kinetic energy e = mc2-y of the particle, (2)

t = e(v E).

This proves that only the electric field exercises power on the particle, as claimed in Example 13 of 7.5.

We shall now describe the motion of a charged particle in some special electromagnetic fields. Consider the case of a vanishing electric field (E = 0).

Equation (2) then implies that the kinetic energy is constant; hence the -y-factor and, in particular, the total velocity v2 are constant, too. The spatial part of the particle's motion e(s) = (x' (s), x2(s), x3(s)) satisfies IldWds112 = v2 = const; hence the reparametrized curve q(s) := o(s/v) is given in its natural parametrization. The Lorentz equation for this curve can be written as

mryv2 = cdd q x B , and it is natural to use Frenet's frame to describe the curve q(s) in euclidean space R3. It is safe again to denote derivatives with respect to the natural parameter s by dots.

Example 3 (Static and homogeneous magnetic field). Suppose that the magnetic field B is static and homogeneous, i. e., constant in space and time. Recall that t, h, b denote the tangent, the principal normal and the binormal vector, respectively, and that c and r are the curvature and the torsion of q. The Lorentz equation is thus geometrically equivalent to e-.

-txB. c

We differentiate this equation with respect to s and use Frenet's formulas for the derivative of h and t:

mryv(id +/c(-Kt+'rb)) =

eKh' C

x B.

9.5. The Lorentz Force

321

Since the vectors t, h', 6 constitute, at every point, an orthonormal frame, we conclude that k = 0. We compute the value of the curvature:

K = 110 =

e jI m-yvc

x BII =

eIIBII mryvc

eIIBII m-yvc

sin(i),B) =

1 - (4.B) 1IIBII

In fact, on can check directly that (4, B) has to be constant, for its derivative is 2 (ii, B); but by the Lorentz equation, ij is perpendicular to B. Alternatively, we could also have derived it from Noether's theorem. In order to determine the curve's torsion, we first compute its binormal vector,

b = Fxh =

xij,

(h x B) =

(h(t'

as well as its derivative, e

b = -K x 11 =

ym vcq x The torsion is then

y mVc e

B) -

B(h,

ryB) h'.

h) _ -e(t' B)

rymvc '

and again constant. From Chapter 5, Exercice 7 we know that a curve whose

torsion and curvature are constant is necessarily a helix, the direction of which is given by B. In the special case that (F, B) = 0, the curve is a circle in the plane perpendicular to B with radius of curvature r and synchrotron frequency w:

r=

1

K

=

mryvc

v

=

eIIBII'

_

r

eliBII rymc

Example 4 (Dirac monopole3). Although their physical existence could never be established, Dirac monopoles have proved important models in theoretical physics. A monopole at the origin of R3 induces the field

B

c

= 2eIIiiII3'1

The Lorentz equation is thus ij

27mvIMII3J7x g

= We are going to prove that the particle moves along a geodesic of a cone with vertex the origin. As in the integration of the geodesic flow or the abstract discussion of Lagrange systems, the proof makes crucial use of a (*)

clever invariant of motion. But the invariant of the Dirac monopole differs from all invariants encountered so far in the fact that it is not linear in ; 3This discussion has been published by Katharina Habermann (formerly Neitzke) in her article K. Neitkze, Die Lorentz-Kraft auf pseudo-Riemannschen Mannigfaltigkeiten, Math. Nachr. 149 (1990), 183-214.

9. Elements of Electrodynamics

322

hence its existence does not follow from Noether's theorem, but only from a direct calculation. First, we observe that (n, 7i) + (n, n) = 0 + 1,

d !n,

because n is perpendicular to 6j by the Lorentz equation (*), and so there exists a constant a E R with (n, rl) = s +a. It also implies that the following derivative vanishes:

0, ds which yields the invariant we were looking for. By the Cauchy-Schwarz inequality, it has to be non-negative; hence there exists a constant k > 0 such that Ilnl12

- (n,

)2 = k2

and

11,7112 = k2+(S+ a)2.

A short calculation for the curvature of the particle's curve of motion yields

K = Itrill =

1117 x ells

=

2TmvJ lot I

1-ln,n>2/1117112

__

sin(n,il) 2'rmvl

In11s

=

1 -oos (2, _ 27mvI InI

111711'-117,17)2

__

k

2 ymvllnl l2

2-tmvl ln113 2'rrmll,II3 If k = 0, the curvature vanishes and the curve is a straight line, i. e., there exist vectors v", w' E R3 such that n(s) = s v + v7. The equation of motion (*) then implies v' x u7 = 0, i. e., v' and 0 are linearly dependent, so thet we finally obtain n(s) = (s + A)t7, A E It , which is the equation of a line through the origin. Let us assume from now on that k > 0. We shall derive the expression for the torsion of the curve. By Frenet's formulas, the principal normal vector hh is

h=-=- k7xil. 1

The binormal vector then becomes

b = ,7xh = -i4x(rlx1), with derivative

=

-kn x (n x ii) =

n

2kmv7II,II3 )) Using Grassmann's identity u7 x (u' x ) = u (v, w-) - v' (u, m'), one shows that 6 x (u" x (i x ii)) = - (u, v3) i x *7, so that the former expression can further dsb

323

9.5. The Lorentz Force

be simplified to dg _

ds

(n, n) 2mvry11i1113

__

(-k r) X rl

2mvryIIi11I3h.

The torsion is thus equal to

tail)

r=

(b, h) _ -2mvryll71113

__

s+a 2mvry

k

+(a+a)23

In particular, the quotient of torsion and curvature satisfies the simple identity r/ic = -(s + a)/k. We prove next that the curve lies on a cone, that is, that there exists a constant vector ii whose angle p with 71 is also constant.

Define the functions

s+a

_ f(s)

2km yv

_

k + (a + a)

'

g(s)

1

2m yv

k + -(s+ a)2

They are primitives of K and -T and have the same quotient, d f (s) f (s) _ k dg(s) r ds = K' ds g(s) Frenet's formulas thus imply immediately that the vector iZ :=

is constant. To compute its angle V with rl, we derive an exact expression for q by applying Grassmann's identity to the binormal, b

1

1

= -krl x (71 x n) _ -k [rl -il(s+a)], q = (s+a)t-k6,

and remark that ii can be rewritten 11= rl/2mkryvllr7l l + h. For the opening angle of the cone we obtain

cos() =

(fi, u')

110 141

-

1

1 + (2mkyv)

9. Elements of Electrodynamics

324

By definition, a curve is a geodesic on a given surface if and only if its principal normal vector is orthogonal to the tangent plane of the surface at any of its points. Denoting by 7P the angle between h and i , we get (h, u)

-

1

Iluli

_

2mkvy 1 + (2mkvy)2

and we see that this is equal to 1 - cos2(cp) = sin(e). Hence, W and V are related through V-V = it/2, and we conclude that h is indeed perpendicular to the tangent plane of the curve. The particle moves along a geodesic of the cone, as claimed.

Exercises

325

Exercises 1 (Kirchhoff Formula). Let u(x, t) be a function defined on JR3 x Ht and S C JR3 a compact domain with smooth boundary. Prove, for each point (Xo, to) E SZ x JR, the Kirchhoff formula:

Jsl

u(xo, to) =

Ou(a,to+

cr 8t

u(1l, to - r/c) or

or(a,to)

- u(a'to

eN

c

r f au (or, to -

dy + J

r

IL

r 8N

8r-1(a,to) C

OM

where r := Ilxo - yII is the distance to the spatial point xo E R3, N is the normal vector to the boundary 00, and = Ox - 1/c28it is the wave operator. Deduce from this the solution formulas for the Cauchy problem of the wave equation by choosing for H a 3-dimensional ball. 2. Under the assumptions of Helmholtz' theorem, the electric and the magnetic field E and B, as well as the current density field J can be written as the sum of a divergence-free and a curl-free vector field,

E = Ediv + Ecuri, B = Bdiv + Bcuri and J = Jdiv + Jcurl a) Prove that the Maxwell equations can then be written as follows: curl(Ed;v) = curl(Bd;Y) =

1

c

-

18Bdiv

8t

8Ed;v

at +

47r

Jai,,,

chv(Bcuri) = 0 div(Ecuri) = 47r LO.

The continuity equation then becomes div(Jcuri) +

OLO

= 0.

Hint: In the proof of Theorem 2, we proved that, under the assumptions made here, any vector field which is at the same time divergence- and curl-free, has to vanish identically. b) On the other hand, for the electric field E we already know the decomposition

E_

c

+grad(-O).

9. Elements of Electrodynamics

326

Prove that this is precisely the Helmholtz decomposition of E if the magnetic potential satisfies the condition called Coulomb gauge:

div(A) = 0. 3. Deduce from Theorem 4 that if Eo(x) and Bo(x) are C3-functions on R3 such that divE0 = divB0 = 0, then the Cauchy initial value problem for an electromagnetic wave

curl(B) = c 5 ,

curl(E)

c at , div(B) = 0, div(E) = 0, E(x,O) = Eo(x), B(x, 0) = Bo(x) has the unique solution E(x, ct)

4act

f f

curl Bo(y)dy +

B(x, ct)

4act

f Eo(y)dy

4ac 8t

,

S2(x,d)

S2(x,d)

curl Eo(y)dy + 41

S2(x,d)

f Bo(y)dy

8t

1S2(x,ct)

4. Using Theorems 1 and 2, determine the electric and the magnetic field in 1R3 which is generated in the following situations:

a) a homogeneously charged ball of radius R with constant charge density P;

b) a charged spherical shell of radius R with constant surface charge density a; c) an infinitely extended straight wire of radius R through which a current with constant current density j flows. 5. Describe the solution of the classical Lorentz equation for a homogeneous, static electric field (E = const) and vanishing magnetic field (B = 0). What happens in the non-relativistic limit of small velocities v << c?

6. The radiation rate of a Lorentz electron is proportional to the second derivative of the vector p(s) E R3,1:

R 211

Compute the general expression for R and discuss the case of planar circular synchrotron radiation.

Exercises

327

7. Show that the Lie algebra o(3,1) of the Lorentz group is not simple-more precisely, that it is the sum of two 3-dimensional ideals. 8 (Infinitely Extended String with Large Oscillations). Consider the Cauchy problem for the one-dimensional wave equation determined by two functions up(x) E C2(R), u1(x) E C1(It):

attu = c2atxU, u(x,0) = uo(x),

t > 0,

atu(x,0) = u1(x)

Part 1. General Shape of the Solution. Prove that there exist two C2functions f and g satisfying

u(x,t) = f(x+ct)+g(x-ct). The solution is hence the superposition of a wave traveling to the left and one traveling to the right. Hint: Introduce the new coordinates xt = x ± ct and show that the wave equation is equivalent to the differential equation a2u

ax+(9x- -

0.

Part 2. Solution of the Cauchy Problem. With the above Ansatz for the solution, the following relations have to hold: uo(x) = f(x) + g(x),

u1 (x) = c(f'(x) - g'(x))

By integration of the second equation over the interval [0, x], prove that the general solution has to be given by

Ed

u(x,t) = 2[uo(x + ct) + ua(x - ct)) + 2cu1(s)ds.

In particular, this argument shows the uniqueness of the solution. How can the result be understood qualitatively by means of the light-cone ?

9. Elements of Electrodynamics

328

9 (One-sided Infinitely Extended Oscillating String). An oscillating string extending infinitely in the positive x-direction is modeled by the one-dimensional wave equation,

attu = c2axxu,

t > 0,

x > 0.

In addition to the initial conditions,

u(x, O) = uo(x), 8tu(x, 0) = u, (x), one has to impose boundary condition which describe the behavior of th "wall" at x = 0 for all times t: u(0, t) = cp(t).

Here uo, 4o E C2(R+) and ul E C'(R+) are required. Prove (using a similar Ansatz as in the preceding exercise) that the solution is x+d

2cjx-d

[uo(x + ct) + uo(x - ct)] + -L

u(x,t) =

2 [uo(x +

ui(s)ds,

d+x

Ct) - up(ct - x)] + p(t - x/C) + ZJ

d-x

ul (s) ds,

where the upper line is to be taken for points (x, t) in the region I, i. e. below

the line x = ct, and the lower line for points (x, t) in the region II, above the line x = ct. What can be said concerning the behavior of the lightcone, in particular in II? Describe, moreover, the regularity properties of the solution on the line x = ct, and explain carefully under which additional conditions it is of class C2 there. x=ct /1

I x

10 (Oscillating String Fixed on Either Side). For the wave equation on bounded domains a separation Ansatz going back to Bernoulli proved to be successful, in that it reduces the problem to one for Fourier series. We are looking for a solution of the one-dimensional wave equation on the interval [0, 1],

attu = c28xxu,

t>0, 0<x<1.

The initial and boundary conditions are

u(x, 0) = uo(x),

atu(x, 0)

= ul (x),

u(0, t)

= 0,

u(1, t)

= 0.

329

Exercises

The explanation for the Fourier method and the related regularity properties to be required of uo and u1 will be treated separately in the next exercise. Start from the Ansatz

u(x, t) = T(t) X(x) . a) Prove that there exists a constant A, such that

X" _ T"

_

X c2T b) Show that the cases A = 0 and A > 0 necessarily lead to the trivial solution u = 0; in the case A < 0 there has to exist a natural number k such that A = -k27r2 and uk(x, t) = Xk(x)Tk(t) = sin kirx (ak sin klrct + bk cos kirct) . The general solution is then obtained as the series 00

u(x, t) _

uk(x, t) _ E sin kirx (ak sin k7rct + bk cos klrct) . k=1

k=1

c) Find an integral formula for the coefficients ak, bk depending on the initial conditions uo and u1. Solution: ak

lore

1

1

2

ul (x) sin k7rx dx,

bk = 2 fo uo(x) sin k7rx dx .

0

11 (Validity of the Fourier Method). By a theorem of Dirichlet, the Fourier series of a function f E C1([O,1]) satisfying f (0) = f (1) = 0 is uniformly convergent and tends pointwise to f. Prove the following lemma:

For a function f E Ck([O,1]) such that f(0), ..., f(k-1) vanish at 0 and 1, there exists a constant A for which the Fourier coefficients of f satisfy the following inequality:

j f(x)sinn7rxdx<

A

Wk

Formulate necessary conditions for uo and u1, under which the solution u(x, t) constructed in the preceding exercise is twice differentiable with respect to x and t and satisfies the initial conditions. 12 (Inhomogeneous Wave Equation).

Part 1. The inhomogeneous wave equation is understood to be the equation (*)

attu = c28x2u + f(x, t)

9. Elements of Electrodynamics

330

with a given function f. We consider this equation on the interval [0, 1] with the initial conditions

u(x,0) = uo(x), i31u(x,0) = ni(x) as well as, first, trivial boundary conditions, u(O,t) = 0,

u(l,t) = 0.

Show that the Ansatz

u(x, t) _

an (t) sin(nirx) n=I

satisfies the boundary conditions, and prove that the functions an(t) are uniquely determined. To do so, prove that they solve an ordinary differential equation of second order, and that the initial values an(0),a'n(0) can be computed from the initial conditions. Hint: The Fourier coefficier..ts of uo, ul and f play an important role here.

Part 2. We are looking for a solution u(x, t) of the inhomogeneous wave equation (*) with arbitrarily given boundary conditions, u(0, t) = spo(t),

u(1, t) = VIM-

Show that the solution of this equation can be reduced to the solution of an inhomogeneous wave equation with trivial boundary conditions (and a different inhomogeneity f!). Hint: Let 4(x, t) be any function in two

variables satisfying 4(0,t) = spo(t) and fi(l, t) = spi(t), e. g. 4-(x, t) _ (1 - x)Wo(t) +xW1(t). Then consider the function v = u - 4D. 13 (Hadamard Example). This exercise intends to illustrate that the Cauchy problem is not posed correctly for every differential equation, i. e., the solution does not necessarily depend "continuously" on the initial conditions. Consider the following Cauchy problem for the Laplace equation: Datermine the solution of the differential equation d2u dt2

+

-dx2= 0 d2u

which at t = 0 satisfy the conditions

u(0,x) = 0, dJu(0.x) = n sinnx, where k and n are positive integers. Suppose, moreover, that x

t>0.

JR and

Exercises

331

a) Verify that this problem has the solution

u(t, x) =

ent - e-nt 2 nk+l

sin nx.

How do (8tu(0, x) I and u(t, x) behave for arbitrarily small t, if n is sufficiently large? b) Now we suppose that we have found the solution 2i(t, x) for the initial conditions

u(0, x) = uo(x), 8tu(0, x) = u1 (x). What is then the solution of the Cauchy problem with the initial conditions U(O, x) = uO(x),

8tu(0, x) = u1(x) +

1

sinnx?

Conclude that the Cauchy problem for the two-dimensional Laplace equation is ill-posed.

Bibliography

Textbooks on Analysis on Manifolds M. P. do Carmo, Differential forms and applications, Universitext, Springer, Berlin, 1994.

H. Flanders, Differential forms. With applications to the physical sciences, Academic Press, New York-London, 1963. 0. Forster, Analysis 3. Integralrechnung im Rn mit Anwendungen, ViewegStudiuYn, Bd. 52, 2. Auflage, Vieweg-Verlag, Braunschweig-Wiesbaden, 1989. K. Maurin, Analysis 11, Reidel/PWN, Warsaw, 1980.

W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, New York, 1976.

M. Spivak, Calculus on manifolds, Addison-Wesley, Reading, MA, 1965.

Textbooks on Differential Geometry W. Blaschke, H. Reichardt, Einfiihrung in die Differentialgeometrie, Grundl. der math. Wiss., Bd. 58, Springer-Verlag, Berlin, 1960.

M. P. do Carmo, Riemannian geometry, Birkhauser, Basel, 1992. M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.

C. Godbillon, Geometrie differentielle et mecanique analytique, Hermann, Paris, 1969.

333

334

Bibliography

A. Gray, Modern differential geometry of curves and surfaces with. Mathematics, 2nd ed., CRC Press, Boca Raton, FL, 1998. M. Spivak, Differential geometry I- V, Publish or Perish Inc., since 1975.

K. Strubecker, Differentialgeometrie I-III, Sammlung Goschen, W. de Gruyter Verlag, Berlin, 1969.

R. Sulanke, P. Wintgen, Differentialgeometrie and Faserbi ndel, VEB Dt. Verlag der Wiss., Berlin, 1972. A. Svec, Global differential geometry of surfaces, VEB Dt. Verlag der Wiss., Berlin, 1981.

Textbooks on Lie Groups and Lie Algebras J. F. Adams, Lectures on Lie groups, Benjamin, New York, and Univ. of Chicago Press, Chicago, IL, 1969.

Th. Brocker, T. torn Dieck, Representations of compact Lie groups, GTM 98, Springer-Verlag, Berlin, 1985.

J. J. Duistermaat, J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin, 2000.

W. Fulton, J. Harris, Representation theory - a first course, GTM 129, Springer, Berlin, 1991.

Textbooks on Symplectic Geometry and Mechanics V. I. Arnold, Mathematical methods of classical mechanics, GTM 60. Springer, Berlin, 1989.

A. T. Fomenko, Symplectic geometry, Advanced Studies in Contemporary Mathematics 5, Gordon and Breach Publ., Amsterdam, 1995. L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol. I: Mechanics, Pergamon Press, Oxford, 3rd corr. ed., 1994. A. Sommerfeld, Lectures on theoretical physics, vol. I. Mechanics, Academic Press, New York, 1952.

W. Thirring, A course in mathematical physics, vol. 1: Classical dynamical systems, Springer, Berlin, 1978.

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Textbooks on Statistical Mechanics and Thermodynamics R. Becker, Theorie der Wdrme, Heidelb. Taschenb. Bd. 10, Springer-Verlag, Berlin, 1985. L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol V. Statistical physics, Pergamon Press, Oxford, 3rd rev. ed., 1996.

A. Sommerfeld, Lectures on theoretical physics, vol. V. Thermodynamics and statistical mechanics, Academic Press, New York, 1956. D. N. Zubarev, V. Morozov, G. Roepke, Statistical mechanics of non-equilibrium processes, vols. 1 and 2, Akademie-Verlag, Berlin, 1996, 1997.

Textbooks on Electrodynamics L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol. II: The classical theory of fields, 4th rev. ed., Butterworth-Heinemann, Oxford, 1996.

A. Sommerfeld, Lectures on theoretical physics, vol. III: Electrodynamics, Academic Press, New York, 1952.

A. Sommerfeld, Partial differential equations in physics, Academic Press, New York, 1949.

W. Thirring, A course in mathematical physics, vol. 2: Classical field theory, Springer, Berlin, 1986.

Symbols .............. 215

J ................ 4, 4,68

GF ................231

0(t2)

* .................7, 7,23

g .................. 209

Wk ...................1

[, ] ..................91

GL(n,K)...........208

*wy ................23

{, ) ................235

23,60 grad(f) .........23,

wi1 .................142

O ..................148

grad(V)............ 255

wo................. 232

V ..................149

I'

.................185

52

.................. 142

0/dt ............... 188 0 ................27, 27,53

h(s) ...............132

R9

HkR(U)

Rf(x,t)............308

0/0xi ..............12

Hk ..................52

curl(V) .............25

El ..................221

R(U, V)W ......... 151

0(f) ....... 24, 65, 101 0(f) ...............306

Aff(K') ............208

............

19

712 .................148 Ik ...................27

................. 208

SL(n, R).. 106, 211, 221 SO(n, K) ......106, 106,221

Ad .................223

I(A) ............... 276 SU(1,1)............227

Ad* ................230

IV .................

ad := Ad........... 224

bid .................151 b(s)

............... 132

Bk(U) ..............

19

ck ...................26

d ................ 16,68

I

305

SU(n) ..............221

................... 148

s-grad(H) .......... 233

.................. 150

supp(ip) ............. 76

11

............ 276

7(p) ............... 275

S(f?, IA)

K*(x) ............. 196

ai ..................141

K(z, M) ............ 37

t( s)

................ 132

c(s) ............... 132

TpRn ................11

5 ...................101

Lg .................208

TXMk ...............54

............ 284

Ev ..................89

TMk ................55

dA, dQ

div(V)........... 24, 24,63

D"(R) .............. 38 dMk ................70 dM2m ............. 229

C(v) ............... 260 L*(wk) .............. 4

T*Mk

............. 230

r(s) ................133

Mk .................47

U(n) ...............221

V(p) ............12, 12,56

dV ...................6

(M2m, w) .......... 229 (M2m, w, H) ....... 236

Ek .................112

mi(f ), i = 1, 2, 3 ... 164 X(M2) ............. 165

Exp. ...............192

µt ..................272

Zk(U)

N(x)

............... 74

Z(B) .............. 278

0,0 ...............69

Zc ................ 223

11,56 f*., .............11,

f*(wk) ..........14, 14,67

vol(Mm) ............

78

.............. 19

0*(F) ............. 231 9ij, 9ii .............. 59 O(n, K) ....... 106, 221 ..................88

337

Index action coordinates, 247 action integral, 259 adiabatic, 290

adjoint minimal surfaces, 171 operator, 101 representation, 223, 224, 230 Aff(K^), 208 affine group, 208, 233, 265 Aharonov-Bohm effect, 297 angle coordinates, 247 angle-preserving map, 108, 181 angular momentum, 268 area-preserving map, 182 Arnold-Liouville theorem, 242 atlas, 48 barometric formula, 282 Bianchi identity first, 186 second, 191

ensemble, 280 symplectic structure, 230 Carathdodory construction, 79

Carnot cycle, 290 catenary, 172, 198 catenoid, 105, 172, 201 Cauchy problem for the Laplace equation, 330 for the wave equation, 305 Cauchy's integral formula, 38 theorem, 37 Cauchy-Riemann equations, 37 Cayley transformation, 227 center, 223 central force, 267 chart, 48 chart transition, 48 Christoffel symbols, 174, 185 Clairaut's theorem, 178, 262 closed, 18

binormal vector, 132 Biot-Savart law, 303

coadjoint representation, 230

Birkhoff's ergodicity theorem, 235 black-body radiation, 293 Bohr-Sommerfeld condition, 248 Boltzmann distribution, 281 Boltzmnann's constant, 285 boundary of a chain, 27 of a manifold, 53 Bour's minimal surface, 201 Brouwer's theorem, 40

Codazzi-Mainardi equation, 153 commutator, 91 completely integrable system, 244 configuration space, 253 conformal group, 315 conformal map, 108, 181 conjugation action, 222 connection form, 142, 186 conservation of energy for Lagrangian systems, 261 for Newtonian systems, 256 in statistical mechanics, 273 conservation of information entropy, 277

canonical coordinates, 236

coclosed, 102

339

Index

340

continuity equation, 296 Coulomb gauge, 326 Coulomb potential, 302 covariant derivative on manifolds, 184, 188, 190 on surfaces, 149, 153 cubic homology group, 30 curl, 25 current density vector, 295

curvature

form, 204 Gaussian, 156 geodesic, 173 lines, 201 mean, 156 normal, 173 of a curve, 132, 198

principal, 167 scalar, 193 sectional, 193 tensor, 151, 186 cylinder, 158, 168, 176, 183

d'Alembert-Lagrange theorem, 258 d'Alembertian, 101 Daniell-Stone functional, 79 Darboux frame, 172 Darboux theorem, 236 Darboux vector, 199 de Rham cohomology, 19 density form, 312 density function, 272 of an electric charge, 295 diffeomorphism group, 88 differential form, 13, 67 closed, 18 enclosed, 102 exact, 1S

harmonic, 102 left-invariant, 210 dimension, 49, 131 Dirac monopole, 321

Dirichlet problem, 95 distribution, 112 integrable, 112 involutive, 113

divergence, 24, 62 geometric interpretation, 91

dual 1-form (of a vector field), 23, 72 Dulong-Petit rule, 289 effective potential, 267 eikonal equation, 311 Einstein equation, 204 Einstein space, 194 energy, 267 for Lagrangian systems, 260

for Newtonian systems, 255 free, 279, 283 inner, 279, 283 kinetic, 254

of a statistical state, 273 of a thermodynamic system, 284 energy density, 316 Enneper's minimal surface, 201 entropy, inner, 279, 283, 284 equation of state, 279 general, 283 equidistribution, 276 equilibrium state, 272 Erlanger Programm, 207 Enter characteristic, 165 Enter equations, 267 Euler-Lagrange equations, 258 exact, 18 exponential map, 192, 212 exterior algebra, 3 derivative, 15, 68

form, I normal vector field, 74 product, 2 Fenchel inequality, 137, 141 field electric, 295 magnetic, 295

field strength form, 312 first Bianchi identity, 186 first fundamental form, 148 first integral, 177, 236, 265 fixed point, 38 fixed point property, 38 flow (of a vector field), 88 form differential, 13 exterior, 1

Frenet formulas, 133 Frenet frame, 132 Frobenius' theorem, 113 fundamental theorem of curve theory, 133

of surface theory, 143, 153 fundamental theorems of thermodynamics, 284 Calerkin's method, 41 y-factor, 319 gas

ideal, 288 real, 293 Gauss equation, 153 Gauss' mean value theorem, 99

Index

theorem, 83 Gauss-Bonnet formula, 164, 179 Gaussian curvature, 156 Gay-Lussac experiment, 292 geodesic curvature, 173 line, 174, 192 spray, 254 Gibbs state, 280 Gibbs' fundamental equation, 284

GL(n,K), 208 gradient, 23. 60 symplectic, 233 graph, 49, 145, 148, 159

Green's formula first, 35, 84 second, 36, 84

group of diffeomorphisms, 88

Hadamard. J., 330 hairy sphere theorem, 82 half-space, 52

Hamilton equations, 234 Hamilton function, 263 Hamilton's theorem, 263 Hamiltonian quaternions, 221, 226 Hamiltonian system, 236 harmonic differential form, 102 harmonic function, 94

Liouville's theorem, 99 maximum principle, 99 mean value theorem, 99 Poisson formula, 99 heat form, 284 hedgehog theorem, 82 helicoid, 172, 201 helix, 131, 176, 198, 266 Helmholtz's theorem, 104, 304 Hessian form, 196 Hilbert, David, 158 Hodge operator, 7, 23 Hodge's theorem, 104, 304 Hodge-Laplace operator, 101 homogeneous space, 220 homotopy, 33, 81 Hopf's theorem, 85 Hopf-Poincare theorem, 166 Huygens' principle, 305, 310

hyperbolic plane, 148, 158, 179, 190, 202, 227 ideal gas, 288

Igel, Satz vom", 82 imaginary part surface, 147, 161 index

of a curve, 199 of a scalar product, 5

341

of a vector field, 166 induced

differential form, 14, 67 exterior form, 4 information, 276 information entropy of a probability measure, 276

of a statistical state, 277 inner entropy, 279, 283, 284 inner product, 4, 68 integrability condition, 114 integral curve, 87 integral manifold, 112 integrating factor, 120 irreversible process, 277 isometry, 180 isotherm, 290 isothermic coordinates, 170 isotropy group, 231 Jacobi, 200

Jacobi identity, 93, 235

Kerr metric, 196 Kirchhoff formula, 325 Kirillov form, 232 Klein, Felix, 207 Lagrange function, 257 Lagrange's theorem, 4 Lagrangian system, 258 hyper-regular, 262 Lambert projection, 183 Lancret's theorem, 134 Laplace-Beltrami operator, 65 Laplacian, 24, 65 Lebesgue measure, 79

left-invariant, 209, 210 Legendre transformation, 260 Levi-Civita connection, 184

Lie algebra, 209 Lie derivative, 89 Lie group, 208 Lie, Sophus, 207 line integral, 30 Liouville form, 230, 265 Liouville's equation, 273 Liouville's theorem for harmonic functions, 99 for symplectic manifolds, 234 Lienard, 309 Lorentz equation, 266 classical, 320 generalized, 317 Lorentz force, 259, 260, 262 Lorentz group, 313

Index

342

manifold flat, 188 orientable, 69 simply connected, 80 symplectic, 229 with boundary, 52

without boundary, 47 Maupertuis-Jacobi principle, 257 Maurer-Cartan equations, 210 Maurer-Cartan form, 210 maximum principle, 99 Maxwell distribution, 281, 292 Maxwell equations in classical formulation, 295 relativistic version, 312 Maxwell relations, 290 Maxwell stress tensor, 315 mean curvature, 156 Mercator projection, 182 microcanonical ensemble, 282 minimal surface, 169 Bour's, 201 Enneper's, 201 Minkowski space, 101, 311 Minkowski-Steiner theorem, 167 modular surface, 146, 159, 201 module, 222 Mobius strip, 51 moment map, 239, 268 momentum, 267 motion, 254 natural parametrization, 131 Neumann problem, 95 Newton potential, 299 Newton's equation, 254 Newtonian system, 254 with potential energy, 255 Noether's theorem, 177, 238, 261 normal curvature vector, 173 normal vector field, 74 O(n, R), 106, 221 orientation induced, 75

of a manifold, 69 of a vector space, 6 osculating plane, 134 Ostrogradski formula, 83 parallel displacement, 189 parallel vector field, 188 parametrized curve, 131 partition function, 278, 283 partition of unity, 76 Peano curve, 129 perihelion precession, 252

Pfaffian system, 112 phase space, 253 Planck's radiation law, 293 Poincar4's lemma, 20, 81 Poincar6's return theorem, 234 point elliptic, 167 flat, 167 hyperbolic, 167 parabolic, 167 umbilic, 167 Poisson bracket, 235 Poisson formula for harmonic functions, 99 for the wave equation, 305 polar coordinates, 57 potential chemical, 286 electric, 297 magnetic, 296 Newton's, 299 retarded, 308 thermodynamic, 289 potential energy, 255 Poynting vector, 316 pressure, 286

principal curvature, 167 principal normal vector, 132 principle of least action, 259 probability current, 275 proper time, 319 pseudo-Riemannian metric, 100 pseudosphere, 107, 158 pullback of a differential form, 14, 67 of an exterior form, 4 quaternions, 221, 226

Rayleigh-Jeans law, 294 real gas, 293 real part surface, 147, 161 representation adjoint, 230

coadjoint, 230, 265 defining, 222 irreducible, 227 of a Lie algebra, 221 of a Lie group, 221 trivial, 222 retarded potential, 308 retarded solutions, 309 reversible process, 277 Ricci tensor, 193 Riemann surface, 106 Riemann, Bernhard, 197 Riemannian curvature tensor, 186

Index

Riemannian metric, 59 scalar curvature, 193 scalar product, 4 Schrudinger equation, 297 Schwarzschild-Eddington metric, 196, 204 second Bianchi identity, 191 second fundamental form, 150 sectional curvature, 193 Shannon, C. E., 276 signature (of a scalar product), 5 simply connected, 80 singular chain. 27 cube, 26 SL(2, R), 106, 211 SL(n, K), 221 slope line, 134 smooth map, 26, 34, 39 solid body, 289 SO(n, R), 106, 221 space form, 196

space of constant sectional curvature, 196 spherical coordinates, 59 spherical mean, 305 spherical pendulum, 249 standard cube, 27 star-shaped set, 19 stationary terminal distribution, 272 statistical state, 271 stereographic projection, 107, 181 Stokes' theorem, 32, 79

343

tangent space of Rn, 11 of a manifold, 54 temperature, 284, 285 tensor product (of representations), 227 Theorems Egregium, 142, 163 thermal efficiency, 291

thermodynamic relations, 290 Toda lattice, 266 torsion, 133, 198 torus, 50 total differential equation, 121

tractrix, 107, 158 transitive action, 220

U(n), 221 unit cube, 26 unit normal vector field, 74

van der Waals equation, 293 vector field, 12, 56 complete, 88 components of, 57 flow of, 88

fundamental, 231 left-invariant, 209 normal, 74 parallel, 188 related, 93 velocity of light, 296 volume, 78 volume form, 6, 70, 229

classical version, 36, 85

structural equations of a Lie group, 210 of a manifold, 187 SU(1,1), 227

wave (electromagnetic), 304 wave equation, 304 Cauchy problem for the, 305 Poisson formula for the, 305 wave function, 297

SU(n), 221

wave operator, 101, 306

support, 76

Weierstrass representation, 171 Weingarten map, 152, 156 Wiechert, 309

of a surface, 133, 142

surface of revolution, 144, 148, 158, 177, 185, 200

Sylvester's theorem, 4 symmetric product, 148 symplectic coordinates, 236 diffeomorphism, 238 form, 229 gradient, 233 manifold, 229 structure, 229 volume form, 229 synchrotron frequency, 321 synchrotron radiation, 326 system, completely integrable, 244 tangent bundle, 55

Wien's law, 294 winding form, 18 winding number, 137, 199 work form, 284 Yang-Mills equation, 316

From a Review of the German Edition: Drawing on his great experience in research, writing books, teaching, and working with

students, Friedrich presents once more a clearly written, smoothly readable self contained textbook The mathematical material and approaches are well motivated, enriched by valuable considerations and reflections. Proofs are elegant, not too tech-

nical and carefully performed ... Each chapter finishes with exercises designed to increase comprehension ... For any student who has passed the linear algebra course and calculus, this book offers on excellent opportunity to learn global analysis and its applications to mathematical physics.

-Mathematical Reviews

This book introduces the reader to the world of differential forms and their uses in geometry, analysis. and mathematical physics. It begins with a few basic topics, partly as review, then moves on to vector analysis on manifolds and the study of curves and surfaces in 3-space. Lie groups and homogeneous spaces are discussed, providing the appropriate framework for introducing symmetry in both mathematical and physical contexts.The final third of the book applies the mathematical ideas to important areas of physics: Hamiltonian mechanics, statistical mechanics, and electrodynamics. There are many classroom-tested exercises and examples with excellent figures throughoutThe book is ideal as a text for a first course in differential geometry, suitable for advanced undergraduates or graduate students in mathematics or physics.

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