Boundary Value Problems for Elliptic Systems

4Pz

(P2

-(Pi

so that Y+ = X+-1. 1

21i ,,.

= i

L-'(;.) di,

0'

.(Iw12

1

20 1-1Q12``0 i 2(1

1012

-

io,

i )

1

Boundary value problems for elliptic systems

64

Then (40) holds when 9 # 0, and, by letting 4p - 0, we see that (40) also holds if (p = 0. Thus (X+, J, Y+) is a y+-spectral triple for LO.). Since

det X. = 0 and det Y+ = -4p/2(1 - j(p12), we see that X. and Y+ are invertible if and only if V # 0. Hence (i) If rp = 0, L(i.) has neither monic y-spectral right divisor nor left divisor. (ii) If cp 96 0, L(A) has a y-spectral right divisor D. - S, where S=X+J+X+1-

-1 and a y-spectral left divisor I1. - S, where Y+1J+Y+= cp

1

Example 3.18 Let a, b, c and d distinct complex numbers and let

L(i) - (1, - a)().

b)

0

1. - a (). - c)(a - d)

Let y be a contour with a and b inside y, and c and d outside y. Then det LO.) has p = 2 zeros inside y. Since

M12=21

J

L-1(2)[I

it]di.=(0

Y

Ob

0

1

S)

and

0

M21 =

2xi

f, (M)

L`(A) d.? = bb

\0 0 / where 6 = -(b - c) -1(b - d)-', we see that M12 has rank 1 and M21 has rank 2. Hence there exists a monic y-spectral left divisor, but no such right divisor. A y-spectral triple for LO.) is given by

X. =(0 0),

J+ = (a

b)'

Y+ = a I

b

(-1

(a Ob)S)

and observe that X+ is not invertible, while Y+ is invertible.

3.8 Changing from complex to real matrix coefficients

A polynomial with coefficients that are matrices with real entries shall be referred to as a real matrix polynomial. In other words, L(i.) _ AJA is a real matrix polynomial if A; = A. Real matrix polynomL(A) for all ials are also characterized by the following condition: %eC.

Monic matrix polynomials

65

Throughout this section we assume that det L(A) 0 0 for real A. Also we let y t be simple, closed contours containing the eigenvalues of LO.) in the

upper and lower half-planes, respectively. Note that if L(A) is a real matrix polynomial and if (X+, T+, Y+) is a y+-spectral triple of L(A), then (X+, T, Y+) is a y-spectral triple of L(A).

In general, if L(A) = Y AJV is a complex p x p matrix polynomial then L(A) = L(A) where L(A) = Y AJA1 and Aj denotes the complex conjugate of A. Let L,(A) and L2(A) be the real and imaginary parts of L(1.), i.e. the unique real p x p matrices such that L(A) = L,(A) + iL2(A). We then associate with L(A) the following real 2p x 2p matrix polynomial: LR(A) =

LE(A)

-L2(A)

L2(A)

LE(A))

There is a close connection between the spectral data of L(A) and that of LR(1.) (see Theorem 3.20). This can be anticipated by examining the differential equation L(d/dt)u(t) = 0. If we write u = u, + iuz, then LR(d/dt)uR(t) = 0,

where UR =

uz! For example, if L(Ao)xo = 0 then the function u(t) = ez°`xo is a solution of

L(d/dt)u = 0 and if we write io = a + ib and xo = x, + ix2 then the corresponding real function (UR(t)

= e"'

xi cos bt - xz sin bt\ J x, sin bt + xz cos bt

xz/ eA01

Iil

Jxo + z ex°'I

it )io

(41)

satisfies LR(d/dt)uR(t) = 0. This implies LR(Ao)xo.R = 0 and LR(7.o)zo,K = 0, where xo,R = ( which is easy to verify directly since LR(Ao)xo.R

L(1.o)xo l _

= -iL(Ao)xo/

lil}xo.

0 0

Thus, to each eigenvalue Ao of L(A) with eigenvector xo, there corresponds two eigenvalues, Ao and J.o, of LR(A) with eigenvectors respectively.

(\

I

) xo and ( if )go,

66

Boundary value problems for elliptic systems

Lemma 3.19 Let A and B be p x p real matrices. Then U)=( I A + iB (B A)( lil lil il)(A and

A - iB)

l

/A

detl

B J = Idet(A + iB)12

Proof The first formula is proved by multiplying out both sides and comparing entries. The second formula follows immediately from the first. By virtue of the lemma we have the following representation: I I 1)(L(A)

I)-'

-il

(42)

L(1.))( -i1 it Since det L(A) s 0 for real A then det LR(A) A 0 for real A. If r (resp. at - r)

it

denotes the number of eigenvalues of LO.) in the upper (resp. lower) half-plane, counting multiplicities, then LR(A) has r + (a - r) = a eigenvalues in the upper half-plane and also at eigenvalues in the lower half-plane.

Theorem 3.20 Let L(A) be a p x p matrix polynomial such that det L(A) # 0 for real A. Let y t be a simple, closed contour containing the eigenvalues of L(A) in the upper and lower half-planes, respectively. Let (Xt, TT, Y±) be a yt-spectral triple of L(A) consisting of matrices, i.e. X+ is a p x r matrix, T+

is rxr, Y+ is r x p, X_ ispx(a-r), T_ is (a-r)x(a-r)and Y_ is (a - r) x p. Then

XR

iI

RT+-CT+

+-\Y+

YR

it

2p x a matrix,

X_

T ) , a x a matrix, 1)

' a x 2p matrix,

I it Y X -iI

is a y+-spectral triple of LR(1.). (I denotes the p x p identity matrix.) Proof We shall verify (1), (ii'), (iii) and (iv) of the definition of spectral triples (see Definition 2.1 and Lemma 2.2). (i) The fact that sp(TR) lies inside y + is clear from the definition of TR .

(ii') Since XtTtY, = 1/(2ni)

AJL-'().)d). for j = 0, 1,... then

X_Tj Y = 1

,-

27ri

A-L(A)d).

1

-2ni ,1

2ni

,.

,i;L-'(i.) dl.

AJL-'(A) d1.

Monic matrix polynomials

67

Note: If y- is traversed in the counterclockwise sense, then y is traversed clockwise. In view of (42) we have

I

jiLR1())d'

I

11

I

)J.i('() it )(x+ T+Y+

)d 1 IY

1

L-

-il

)

I

I

it J

-'

= X R(TR )'YR

forj=0,1..... (iii) From the definition of XR and TR, one obtains the following:

col{X (TR )'}j=o =

(43)

X+T+'

X_T'='

-iX+T+' iX_T'-' By means of block-row operations (i times the first block of p rows, added to the second block of p rows, etc.), the matrix (43) is transformed to

(44)

X+T+' 0

X_T'=' 2iX_

Since col(X+ T+) j:o has r linearly independent columns and col(X_ T'_) j=o has a - r linearly independent columns, it is clear from the form of (44) that col{XR(TR)J}j=o has a linearly independent columns. (iv) Similarly, one proves that row{YR(TR )'}j=o has a linearly independent rows.

Before stating the next results, some notation is required. For k = 1, 2, .. . and n' = 1, 2, ... , let (as in §3.5)

Boundary value problems for elliptic systems

68

I Mkm

= 2ni J

L-'(7.)[I. .Am-'1] d1,,

pk x pm matrix

Ak_ll

Y

and I

Mm

tai

J

LR`().)[I

2pk x 2pm matrix.

A"'-'1] dA,

Also define Af,,, just as Mkm but with L replaced by L. In the definitions of Mkm and Mkm, the matrix I is the p x p identity while in MR , it denotes the 2p x 2p identity. In view of (42), we have

tai f".

1.1LR'(i.) dA

('iI %} - ( IiI il)(C'

C';}

for j = 0, 1, ... where

CJ=

-'f, AJL-'(A) dA, 2a! .

('

C= 2ai 1J

.

A E 1(A) dA

Hence it follows that M mQm = Qt

Mkm (s)

Mkm

where

r1

1

QA =

/l

1

- if k blocks

it k bl cs

a 2pk x 2pk matrix. Here I is the p x p identity and there are k blocks of I

and k blocks of IiI In general, Mkm # Mk, (e.g. if L(A) = (A - i)2 then M12 # M12); however,

G)

we claim that Mkm = Mkm whenever L(1.) is monic of degree I and k + in < 1.

Monic matrix polynomials

69

To prove this, first note that if k + m < 1, then the blocks of Mk,,, _ [C

1-0 j=O have the form

Ci=2ni1

y.

1.'L-'().) d1..

where j < I - 2.

If L(i.) is monic of degree I then ).jL-'(i.) is of order no greater than 2-2 as JAl

oo, and it fo llo ws that

).JL-'(A) di = I J

J.1L-'().) dA

JJJ

and

J

1.'L-'(i.) d.?

for j = 0, ... ,1- 2. The claim now follows immediately. Lemma 3.21 Suppose that L(A) is monic. Then the following identity holds for any positive integers such that 2s <, 1:

Mu=Q.1 M. M )Qs'

\

u

Therefore det MR = Idet MUI2, and det MR # 0 if and only if det M # 0.

Proof The blocks in the matrix M have the form 1

2iri .,.

A L-'().) d1.,

where j < 2s - 2 < I - 2. Hence, by the remarks above, it follows that M,, = M. The corollary now follows immediately from (*). Let L(i.) be monic and suppose L(1.) = L2(2) L,(i.) where the right divisor L1().) has invertible leading coefficient. Then the left divisor L2(A) must also have invertible leading coefficient. Combined with Theorems 3.14 and 3.16 this observation leads to the following theorem.

Theorem 3.22 Let L(A) be a monic p x p matrix polynomial of degree I = 2s

such that det L(2) # 0 for real d, and suppose that det L(A) has exactly ps zeros in the upper half-plane, counting multiplicities. Let y+ be, as usual, a contour in the upper half-plane containing the zeros of det L(A) there. Then the following statements are equivalent:

(a) L(1.) admits both a right and a left monic y+-spectral divisor of degree s;

(b) det M # 0; (c) Both A.2, and A.2, have rank ps; (d) LR(2) admits a monic y+-spectral right divisor of degree s.

70

Boundary value problems for elliptic systems

Proof The fact that (a) a (b) is known from Theorem 3.16. (a) ,c:> (c). This follows from Theorem 3.14, since by complex conjugation

it is clear that L(;.) has a monic 7'-spectral right (left) factorization if and only if L(;.) has a y+-spectral left (right) factorization. (b) - (d). Since LR(i.) is a real matrix polynomial, it follows by complex conjugation that LR(i.) has a monic y+-spectral right divisor of degree s if and only it has a monic y'-spectral left divisor of degree s. Therefore, by Theorem 3.16, (d) holds if and only if det M R A 0. (In view of (42), LR(;.) has exactly 2ps eigenvalues in the upper half-plane, counting multiplicities).) The

fact that (b) and (d) are equivalent is now clear from Lemma 3.21.

Corollary If L(;.) is a real matrix polynomial satisfying the hypotheses of Theorem 3.22, then M3 2., has rank ps if and only if det M. -A 0.

Proof By Theorem 3.14, Ms.25 has rank ps if and only if L(;.) has a monic y +-spectral right divisor of degree s, i.e. L(;.) = L2(;)L,(;.) where sp(L,) lies in the upper half-plane and sp(L2) lies in the lower half-plane. However, by complex conjugation it is clear that this holds if and only if L(;) = L2(1)Li(U) where sp(L2) = sp(L2) lies in the upper half-plane and sp(L,) = sp(L,) lies in the lower half-plane. This means that L(i) has a monic y+-spectral right divisor of degree s if and only if it has a monic y'-spectral left divisor of degree s. Hence, the corollary follows from Theorem 3.16. We conclude this section by showing in detail the connection between y+-spectral factorizations of L(;.) and LR(;.).

Suppose that L(;.) admits both a right and a left monic ;+-spectral M+ factorization, i.e. L(;.) = L+ and M-) are monic of degree s with spectrum contained in the upper (resp. lower) half-plane. In view of (42) I

L-(;.)

1

(-il

il)(

L+(i.)

1

1

il)

M+(;.))(

''

(45)

so we obtain the monic y+-spectral right factorization

A(;.) -B-(i.)l A+(i!) -B(A) B+(;) A+(;) A_() J

LR(;) _ (B_(,)

(46)

where the matrix polynomials A * (1.) and B f (;.) are defined such that LT (A) = A+(A) + iB+(;.),

L-(;.) = A-(;.) + iB-()),

M'(;.) = A+(i.) - iB+(i.), M+U.) = A-(A) - iB-().)

To justify this, note that (46) is obtained from (45) by means of Lemma 3.19, and since det

B+(A +(A) )

= det L+(;.) det M-(i.),

Monic matrix polynomials

71

with a similar result for the other factor of (46), it follows that the factorization (46) is a y+-spectral right factorization of LR(A). Also, in view

of the fact that L'().) and M±(i.) are monic of degree s, Af(A) is monic of degree s and B±(;_) has degree less than s. Hence both factors in (46) are monic of degree s.

These steps are also reversible, i.e. given a y+-spectral factorization of LR(i.) of form (46) one obtains right and left y+-spectral factorizations of L(i.).

4 Further results

We consider the spectrum at infinity and related results in §§4.1 and 4.2. The main result is Theorem 4.3 which shows how the finite and infinite spectral

pairs together form a generalization of the concept of a standard pair for the case of non-monic matrix polynomials. This result is not needed in the sequel, except very briefly in §10.4, but it is included here for the sake of completeness. In §4.4 we consider a product of matrix polynomials, L(A) = L2(A)L1(A)

and show how to obtain a y-spectral triple for it in terms of y-spectral triples of each factor. The main result is proved in Theorem 4.17. Let (XI" T,, Y; ) be a y-spectral triple of L,{i.) and denote the base space by ¶W, (j = 1, 2). Then the product L(A) = L2(A)L,(1.) has the following y-spectral triple:

X+=[Xi A],

T+

0'

Y'TX2 2

Y+_ 2

where

A=

B=

1

- L-'(i.)XZ (U. - T2 )'' di. 2iri 1

I

2iri ,

Tl)-'Yl{LZ'(A) -XZ(IJ.-T2)-'YZ}dA.

This theorem is proved using the operators Q and S which are defined in the statement of Lemmas 4.12 and 4.15. In §4.5 we look at how these operators Q and S are affected by a Mobius transformation of the matrix polynomials. Section 4.3 is essentially a continuation of §2.2 and is needed for §§4.4 and 4.5.

In Part V of the book we will prove an index formula for elliptic systems in the plane based on the spectral theory of matrix polynomials which has been developed in Chapters 1 to 3. Sections 4.4 and 4.5 develop further tools which are important for a full understanding of the index formula. 72

Further results

73

4.1 The inhomogeneous equation L(d/dt)u = f Let LO.) = Ei=o Aj2j be a p x p matrix polynomial with det L(2) # 0. The Laurent series for the resolvent

L-'(1.) = i Cj)j j= - 30

converges when JAI is large, where

j = 0, ±1, ±2,... (1) 2ai,fr and r is a large circle having all of sp(L) in its interior. Also we assume Cj = 0 when j > K and K is the smallest non-negative integer with this Cj

property, i.e. K is the order of the pole at infinity. For j < - I the coefficients Cj can be expressed in terms of a finite spectral triple of L(A), i.e. Cj = X T-j-' Y where (X, T, Y) is a r -spectral triple of L(A). By analytic continuation we have

L-'(}) = X(12 - T)-'Y+ E Cj)' j=o

(2)

since both sides have the same Laurent expansion when JAS is large.

(To compute the coefficients when j >, 0 we need to introduce spectral triples at infinity. This will be taken up in §4.2.)

The representation (2) of the resolvent L-'(2) can be used to obtain a particular solution of the equation L(d/dt)u = f. Theorem 4.1 Let L(A) be a p x p matrix polynomial with det L(2) 0- 0. Then

for any f e C'(R, C°), the differential equation L(d/dt)u = f has the particular solution u(t)

= 2I7ri ,1 ot (fr

e

uL(A) dA f(s) ds + j Cjf'j)(t) J=O

where r is a simple, closed contour having all of sp(L) in its interior, and Cj are as defined above.

Proof Let (X, T, Y) be a finite spectral triple for L(A), and let

2I fo, \f reuL(A)dAf(s)ds

11°(t)

_

J`

X e(t-.+)r Yf(s) ds

0

We have to show that L(d/dt)(uo(t)

+j0i Cjf'h(t)) = f(t)

(3)

Boundary value problems for elliptic systems

74

First of all, it follows by induction that

d

J

XT'e"-')T Yf(s)ds

0

Then,

L(d/dt)uo =

_

,=o

t=,

+

A,

d' dt

uo

A,(X Yf rr 1'(t)

r Jo

+

+X

Yf(t))

Y A,XT' 0- -'1' Yf(s) ds 1=o

(Aj+1XY+...+A,XT'-j-'Y)

Y j0

djf(t),

where, in the last equality, we used the fact that Yr=o A,XT' = 0. By Lemma 4.2 (see below), it follows that 1-, L(d/dt)uo = Y- Li+1(d/dt)XTTYf(t) (4) j=o Now multiply both sides of (2) by L(1.) on the left to obtain

I =L(A)X(IA-T)-'Y+L(A) i CiAi, j=o

and then it follows from Lemma 1.18 that

/-I

K

Lj+,(A)XTi Y =I - L(A) E Cj1.j

(5)

j=o j=o Finally, replacing A by d/dt in (5) and using (4), we obtain (3).

Lemma 4.2 Let L(A) be a p x p matrix polynomial and define matrix polynomials L,{;.) (j = 1, ... , 1) as in Lemma 1.18. Then for any admissible pair (X, T), the following identity holds: 1-1

1-1

Y- Li,,(A)XTV = Y (AJ+1X +... + A,XT'-i-')AJ.

j=o

i=o

Proof Let .t be the matrix 2.1(5). Then 1-1

Y Lj+1(A)XT' = [L1(A)...L,(A)]'col(XTj)j'=o j=o

_ [I _

r-1 i=o

(Ai+IX +...+A,XT'-j-')Aj.

Further results

75

4.2 Infinite spectral triples AjA1 be a p x p matrix polynomial with det L(A) # 0, and Let L(A) let a denote the degree of det L(A). We have a < pl, and a = pt only when

leading coefficient, A,, of L(A) is invertible. Now consider the matrix polynomial L(A) =

A'L(A-') _ i

A,-1A'.

1=0

If a < pl then det L(A) has a zero of multiplicity pl - a at the origin. An infinite spectral triple for L(A) is by definition a spectral triple for L(A) corresponding to the eigenvalue 0. The base space of an infinite spectral triple (Xm, T,,, Yom) for L(A) thus has dimension pl - a. By making the substitution A -+ A-1 we have 121ri

dA

fi

1f 2zti

r

j = 0, 1....

A1-2 -JL- '(A) d).

(6)

where y is a small circle centered at the origin not containing any other zeros of det L(A), and r is a large circle having all of sp(L) in its interior. In view of (1) it follows that 1+1Y.

when j>0.

Since T is nilpotent (eigenvalue 0), (1-

1

(7)

= I + AT, + AZT , +

and we have therefore proved

L-'(A) = X(lA- T)-1Y+

AT.)-1Y

(8)

Remark Since C1= 0 when j > K then 7." = 0. This follows from (7) due to injectivity of and surjectivity of row(T;, Y. ),J-=01. We know from §2.2 that col(XT1)js, has left inverse The following theorem shows more, namely that col(XT1)'J=O -' is part of a larger matrix which is invertible. Theorem 4.3 Let (X, T) and (X,,,

be finite and infinite spectral pairs for

L(A), respectively. Then the matrix

X S1-1

X is invertible.

Boundary value problems for elliptic systems

76

Proof Note that S,_, is a square pl x pl matrix since the base space of (X, T) has dimension pl - a. has dimension a while the base space of (Xx, Suppose first that 0 # sp(L). That means det AO # 0 so the matrix polyhas pl eigenvalues (counting multiplicities). We nomial LO.) claim that

X = [X

T=

X,.,C],

T-' Tx

is a standard pair for E(i.). In fact, Theorem 2.21 implies that (X, T-') is a spectral pair for L corresponding to the eigenvalues in sp(L)\O, and by

definition (X., Tx) is a spectral pair for L at 0, so the claim follows immediately from Lemma 1.23. Hence col(XTJ);=o is invertible. Since

'=S

col(XT'-1

.

T-u- n

then S, _ 1 is also invertible.

In the general case we choose a e C such that the matrix polynomial L(i.) = L(i. - a) is invertible at A = 0. By virtue of Theorem 2.21, L(A) has a finite spectral pair

where t= T + al

(X, T),

and an infinite spectral pair

(X., T.),

where T. = T.(1 + aT_)-'

Thus 91, , = col(XTJ S

l

J=O

is invertible. Finally observe that

col(X(T + al )J X. T'' J(1 + aTr)J);=o

u-

= M = [M1 ] and Mj.t = 0, j < k, Mj, =

()a'_i.

k when

j, k = 0, ... , I - 1. Hence S,is invertible. There exist operators Y and Y,o such that (X, T, Y) and (X., T, Y,,) are finite and infinite spectral triples of L(A). We can give an alternative proof of Theorem 4.3 by finding an explicit formula for the inverse of S,_,. Theorem 4.3' 1-1

[Y...Tl-'Y]2' ) C [T.

(9)

where l" is defined by equation (5) in §2.1 and the matrix 2' is defined by equation (10) below.

Further results

77

Proof Let

j = i,..., l

L,{T, Y):=iTk-iYAk, k=j

(note the correspondence with the notation L,{3.) used in Lemma 1.18). Then we have

XT"'Lj(T, Y) =

v=0,...,1- 1 and j =

21ri

i

k=j

Jr,

Akik d).,

L Also let

Lj(TT, Y.):=

k=j

T -jYYAg-k

(A,_k are the coefficients of L(i.)). Then we have XWTs Lj(Ta, Yom) =

_

1 2n!

i"-ji-1(j) Y '4,-kAk dt k=j

7

; -"+i-ZL-!(i.) Y Akik d;

2ni r

k=o

(for the second equality see (6)). Hence

Y) + X.

T`".Ll-j(T0, Y.)

f 2ni r

=

2ni fr

k= j+,

1 J r1 )."-j-'I d).

A"-i- iL-!(i.)

Akik d1. k=o

2iri

6,j I,

v=0,...,1-1 and j=0,...,1-1

Thus

L,(T, Y)

L,(n T ,Y

.

L,(T, Y)

x)...L,(Tx. Yom)

and this proves (9) once we note that

[L,(TT, Y.)...L,(T, , Y.)] _ [T;, I Ys... Y. where

Y, = vla-! =

(10)

Boundary value problems for elliptic systems

78

Corollary 4.4 Let

X

[Y... T'-' Y]Y

Pr =

XT' and

X. T.-' P. _

[Tz' yam... Yom]., X,

where -JY' is defined by equation (10). Then Pr and P,, are projectors and w have

(a) Pr + P. = I,, (b) Pr P. = P.Pr = 0

Proof That Pr is a projector is known from §2.2. That P., is a projector follows from the equation P,, = aP,,a-', where P,, is the projector for the matrix polynomial L(i.) corresponding to the contour and a is the

permutation of p x p blocks in C" that maps the j th block to the

(I - j + 1) th block, j = 1, ... , !. (Note that Y' = a 2a -', where I is the matrix 2.2(5) for the matrix polynomial L(i.).) The fact that Pr + Pte, = lo, is simply another way of expressing the equation (9). To prove (b) note that (9) also implies

x

[Y...T' Y].T C[T=

Y....Y.IXI

X,,,T' '

_

-'

1.M

0

0

lye

XT X, where IJl and M., are the base spaces of (X, T, Y) and (X,,, T, Y,,), respectively. Hence

X

[T;' Y.... YA1,

=0

(11)

XT'-and

X T'' [Y...T'-'Y]X

=0.

(12)

x' From (12) it follows that Pr P,, = 0 and (11) implies Ps,Pr = 0. The following proposition will be of use in Chapter 10. In the statement of the proposition it is assumed that (X, T, Y) and (X., TT, YJ are matrices (rather than operators).

Further results

79

Proposition 4.5 Let r be a positive integer. Then for any r x a matrix, M, and

r x (pl - x) matrix, Mx,, there is a unique r x p matrix polynomial B(i.) _ D=oBjAJ such that

Y B;XT' = M

(13)

j=0 and

-1 j=0

BjX,Tx j-' = M.,

(14)

given by 1-j-1

1

Bj= M Y T"YAJ+k+1 + M. - L T.YYAj_k k=0

(15)

k=0

forj=0,....1-1. Proof The equations (14), (13) can be written in the form [BO...B,_1]S1-1

= [MM.],

which has the unique solution [BO...B1-1J = [M Mx]SS1-11

=[M MA

IT...T'Y]. (IV; 1Y. .Y.1-TV

Upon substituting the definitions of f and Z' we obtain the formulas (15) for Bj.

Remark Since [Y. T1-' Y]2" is a left inverse of col(XTJ)' one solution of (13) is given by Bj = 1 TkYAj+k+1, j = 0,...,1- 1. This corresponds to choosing M,, = 0. Addendum: infinite spectral triples and linearization

We now show that a linearization of a matrix polynomial L(A) (see §3.2) may be obtained from the spectral data as given in finite and infinite spectral pairs of L(1.). This result is then used to prove a generalization of Theorem 3.2 for the non-monic case (Theorem 4.7). In the statement of the following theorem, S1_ 1 denotes the p1 x pt matrix

defined in Theorem 4.3, and S1_2 denotes the following p(l - 1) x pl sub-matrix

X

X,, T1"

Si-2= XT'-2

X1

2

80

Boundary value problems for elliptic systems

Also recall the definition of the companion polynomial CL(7.) from §3.1:

i.+I

CL(;) =

(16) 0

AO

A,

-1 A,-,

Theorem 4.6 Let (X, T) and (X., TTY) be finite and infinite spectral pairs of L(i.). Then

CL(;)S,-, _

(s2)(IA - T T i. -

(17)

I),

where W = [A, XT' -1 -J;=a A,X.TX ' -']. Proof By equating like coefficients of i. on both sides of (17), we see that (17) is equivalent to the following pair of equalities:

I

S,-, _ (S -,)\/ 1+1

I

(18)

Tom)

A,

and

r0

1

-0 1

-1

II

W \A0

I

A,

But, equality in the first 1- I block-rows of (18) or (19) clearly holds for any admissible pairs (X, T) and (X.,,, T,,,), and in the last block-row it holds since (in fact, if and only if)

I A;XTT=0,

1=0

E A;Xx7,i=0. j=o

Corollary The matrix CS-W2) is invertible.

(20)

Further results

81

Proof In view of Theorem 3.5, sp(CL) = sp(L). Also sp(T) = sp(L). If we choose A 0 sp(L) then the equation (17) implies that I T 1 .), J and S,_1 are invertible. CL(1CIA

S- I is invertible since

Tmt-1

Recall that a triple of operators (X, T, Y) is called an admissible triple if X e 1(C P, TI), T e 2'(!R) and Y e 2(911, C P) for some finite dimensional vector space 9R. If dim 91 = p, we say that (X, T, Y) is an admissible triple of order p. Theorem 4.7 Let L(A) = D=o AJA' be a p x p matrix polynomial of degree I. Let (X, T, Y) and (X,,, T., Ym) be admissible triples of order a and pt - a,

respectively, where a is the degree of det L(1.). Then the following are equivalent:

(a) (X, T, Y) is a finite spectral triple of LO.) and (X., Tom, Ym) is an infinite spectral triple of L(A), (b)

X(1A - T)-' Y+

YW,

when j=0, 1,...,1-2,

(c) (X, T) is a finite spectral pair of L(A), (X,,,

{0} and

is an infinite spectral pair

of L(A), and

(_Y

[0...01]T,

Yy) = (SWW)

where W is the matrix defined in Theorem 4.6.

Proof (a) . (b) This was proved in the discussion at the beginning of §4.1. (b) (a) Now suppose that (b) holds. Since {0}, then Tkk,, = 0 for some k. Hence (21)

which is a polynomial in 1, and it follows that L-'(A) - X(IA - T)-' Y is analytic in the finite complex plane. Hence

1

2rti J r

i.'L-'(A)di.= 1 VX(I).-T)Yd1. 21ri

r

= X T' Y

(22)

when j = 0, 1, . . . , and it follows that

12ai J r ('

col(XT')!=o row(T'Y);=o

= Pr-

1

L-'(A)[I...1.`-'I)Y dA

Boundary value problems for elliptic systems

82

Since Pr has rank a and the base space of (X, T, Y) has dimension a, it follows that col(XTJ)!-.o and row(TJ Y)f;a must be injective and surjective, respectively. Hence (X, T, Y) is a finite spectral triple of L(J.). We now show that L-'(A) - X (IA Y has an analytic continuation near A = 0, where L(A) Indeed, by making the substitution

w = A`, we obtain

1Jr 2ai

2ni

Y

1 f WI-J-2L-'(w) dw 2ai Jr

= XT'-z-Jy V. when j = 0, . (.' . ,1- 2, while for j

12ni J I

(l - 1 + k (k >, 0) we have

AJL-'(A) dA =1 J ,lk-1L-'(A-') d A 2ai I

2ni

+

,

1.kX(I -AT)-'YdA

1 2ai

T.)

f"

Y. dA

X. P. Y. As usual, y" is a small circle centered at the origin, of radius < II TII -'. In the

above calculation, we used the fact that (I - AT)` is analytic when Al 1< II TII; , and (1). -

is analytic when A # 0.

Thus L- (A) - X,jIA - T.)-'Y,, has an analytic continuation near A = 0. By the same method as shown for the finite spectral triple, it follows that (X,,, T, is a spectral triple of L'(2) corresponding to A = 0, and hence an infinite spectral triple of L(A). (a) (c) Suppose that (a) holds. Then trivially (X, T) and (X,,, T,p) are finite

and infinite spectral pairs of L(A), respectively. In view of (6) we have

X.P. Y.=XT'-'-JYwhen j=0,...,1-2 and 1-1

A,XT'-'Y+ Y.

J=o

-Jy 2ai

A,

+-

r

2ai J=0

AJ

A-'

I

2ai

1.'-'L-'(A)dA

('r

AJ-'L-'(A)d2

r i AJA' L- '(A) d1. J=O

=1 J A-'I dA=1. 2ni r

83

Further results

It follows that CSt-2

W

(23) X-Y.)

\1/ (a) Suppose that (c) holds. Then there exists an operator Y' such that (X, T, Y') is a finite spectral triple of L(A) and an operator Y. such that (X., T,, Y,) is an infinite spectral triple of L(A). Now since (23) holds for Y' and Y. then (c)

0

Y \ = 1(s,_,\ I Y' 1 = W) 111 \-Y'J" 1

1

so that Y = Y' and Y. = Y., and (X, T, Y) and (X., T,,,

are finite and

infinite spectral triples of L(A), respectively.

43 More on restrictions of spectral pairs This section is essentially a continuation of §2.2. In the following theorem L,(A) and L(A) are any p x p matrix polynomials and y is a simple, closed contour not intersecting sp(L1) or sp(L). AJAJ. If (X+, T+) and (X', Ti) are y-spectral Theorem 4.8 Let L(A) pairs of L(A) and L,(A), respectively, the following are equivalent:

(a) The rational matrix function L(1.)LI '(A) has an analytic continuation inside y;

(b) F1=0 AJXi (T, )J = 0; (c) (X', Tl) is a restriction of (X+, T+). Proof Let Y3 be the unique operator such that (X +, T; , Y;) is a y-spectral triple of L1(1.).

(a) a (b) L(J.)L! '(A) has an analytic continuation inside y if and only

l/2ni L A'L(A)L '(A) d1. vanish for k = 0, 1..... In view of property (ii') of spectral triples these integrals are equal to )kY; and due to property (iv) they vanish exactly F;.o AJX; if the integrals

when > .,oAJX+(T,+)J=0. (b) a (c) The equation Fi=o AJX, (T1)J = 0 holds exactly when (X', Tl ) is a partial y-spectral pair of L(A), which in turn holds if and only if (Xi , T,+) is a restriction of (X+, T+).

Corollary 4.9 L(A)L1 '(A) is analytic and invertible inside y if and only if the 7-spectral pairs of L(A) and L1().) coincide.

84

Boundary value problems for elliptic systems

Proof Applying Theorem 4.8 to the rational matrix functions Q().) _ L(i.)Land Q-'(i.) we see that (X', T;) is a restriction of (X+, T+) and vice versa. It follows that (X', T,) and (X+, T+) are similar.

Recall that (c) means that there is a T+-invariant subspace 2 such that (X+12, T+IP) is a y-spectral pair of L,Q). This holds if and only if every u e 9R'' has a representation u(t) = X+ e`T' c for some c. Hence (a), (b) and (c) are also equivalent to the following condition: Ali, c T Note that 2 is unique due to the injectivity of col(XT+)j=o Corollary 4.10 Let (X+, T+) be a y-spectral pair of L(;.), and denote its base space by 9R+. If L(.)L1 '().) has an analytic continuation inside y then the unique T+-invariant subspace 2 such that (X112, T+Ii) is a y-spectral pair of L1(1.) is

2 = {c; c e 9)1+, L,(d/dt)X+ e`T' c = 0},

(24)

Proof The inclusion c is clear. Conversely, if we have c e U1+ such that L,(d/dt)X+ e`T' c - 0 then the function u(t) = X+ e"c belongs to TL'. But (X+12, T+I e) is a y-spectral pair of L,(;.), so u(t) = X+12 ear+I2 c', for some c' e 2, and thus X+ e`T' c = X+ e`T ` c' for all t e R. By the injectivity of col(X+ T. ); =o it follows that c = c' e 2.

Remark We can also describe 2 as the set of all c e At+ such that It

forallk=0,1,...

Y J=o

where A;", j = 0, 1, ... , l,, denote the coefficients of L1(1.). We mention without proof the analogue of Theorem 4.8 for left y-spectral pairs, where y is a simple, closed contour not intersecting sp(L) or sp(L2).

Theorem 4.11 Let L(i.) = E-=o Aj.it. If (T2+, YZ) and (T+, Y+) are left y-spectral pairs of L2(A) and L(1.), respectively, the following are equivalent:

(1) The rational matrix function LZ '(i) L(i.) is analytic inside y;

(2) E. =o(T2')'YzAj=0; (3) (742 +, YZ) is a restriction of (T+, Y+) (see Definition 2.16).

Corollary LZis analytic and invertible inside y if and only if the left y-spectral pairs of L(i.) and L20.) coincide.

4.4 Spectral triples of products

In this section we construct a y-spectral triple for a product L(i.) _ L2(i.)L,(A) of matrix polynomials given a y-spectral triple for each factor. We do this using the operators Q and S which are defined below in Lemmas 4.12 and 4.15.

Further results

85

Lemma 4.12 Let L(i.) = L2(i.)L1().) be a product of matrix polynomials. If (X+, T+), (XZ , T2+) are y-spectral pairs of Lu), L2(;,), respectively, then there is a (unique) surjection Q E x(9)1+, 9112) such that L1(d/dt)X+ eIT+ = XeIT2' .Q (25)

Further, it follows from (25) that ker Q = 2, where 2 is the unique T+-invariant subspace (24) such that (X+I,,, T+Ie) is a y-spectral pair of L1(i.).

Proof Let the base spaces of (X+, T+), (XZ, T2+) be denoted

911+,

9112 ,

respectively. For any c E M+ the function v = e'T' c satisfies the equation L2(d/dt)v = 0 and hence belongs to :U1 2. It follows that for every c c- 9J1+ there exists a unique c2 E 911; such that L1(d/dt)X+ e'T. c XZ erT, c2 (26) Now define Qc = c2; then Q is a map from `xli+ to 9112 such that (25) holds. To show that Q is surjective let c2 E 9112. By virtue of Proposition 2.5 the function v = X2 e'T2 c2 has the form

e''`L2'(i)p(1)d1.

V=-1_

2ni for some polynomial p(A) whence v = L1(d/dt)u where u =

12ni e" L-'(i.)p(i.) di.

Now u c- 9R2, so there exists c e 9)t+ such that u = X. e'T' c. This means that (26) holds and Q is surjective. Due to the injectivity of col(X2(T2)')"= o we see that Qc = 0 if and only if L1(d/dt)X+ e'T' c = 0, which, in view of (24). holds if and only if c e 2. Hence ker Q = Q. Lemma 4.13 In addition to the hypotheses of Lemma 4.12, let Y+ and Y,' be the unique operators such that (X+, T+, Y+) and (X2'Z , T2, YZ) are y-spectral triples of L(A) and L2(i.), respectively. Then

QT+ = TZQ and QY+ = YZ. Proof Differentiating both sides of (25) with respect to t gives X2 e'r2 T2 Q = L1(d/dt)X+ etr+ T+ = X2 e'TI QT Since co1(X2 (T,)')l= .' is injective we obtain T2 Q = QT+. Similarly, X2+ e'T2

QY+ = L1(d/dt)X. e'T' Y+

e'"L-'(i.) di.

= L1(d/dt) 27ri J. 1

2ni

e'" L- 10.) dI.

=X+e'T' y;, 2 so that Q Y+ = Y2.

86

Boundary value problems for elliptic systems

Remark 4.14 Let it denote the canonical projection 912+ 9R+/2. Since 2 = ker Q there exists an operator Q e 2'(98+/Q, 9R2') such that Q = Qtr. Also let T+, Y+ be the operators induced by it from T+, Y, respectively. In view of the preceding lemma we have and

QT+ = T1

QY, = Y,+.

Since Q is invertible this means that (Ti, Y') is a restriction of (T+, Y,) (see Definition 2.16).

Remark Lemmas 4.12, 4.13 are essentially a further development of the remark made after Definition 2.16. As we showed in Lemma 4.12 the map Q is surjective. The next lemma shows that there is a natural way to define a section of Q, that is, an operator

S E 2'(9R', 9R+) such that QS = I (the identity operator on M' ). Lemma 4.15 Let the hypotheses and notation be as in Lemma 4.12. There 'W4) such that exists a unique map Sc X+ e(T. S =

1

2ni f,

eu Li'(1)Xz (I;. - T)-' dl.,

(27)

for all t E ff8, and the equation (27) implies that QS = 1.

Proof Let c2 E 9R' and consider the function v(t)

2ai f e`'' Li '(2)X2 (11. - T2+ )'c2 di.. s

Since L2(i.)X2 (Ii. - T2*)-'c2 is a polynomial in i. due to Lemma 1.18, we see that L(d/dt)v = 0, so v = 9RL , and therefore v(t) = X+ e` c for a unique c e W. Now define Sc2 = c; then S is a map from 911' to 9R+ such that (27) holds. The uniqueness of the map S follows from the injectivity of col(X+T+)J'=o To show that S is a section of Q, observe that differentiation of both sides of (27) with respect to t yields Ll(d/dt)X+ e'T. S =

1

2Ici

,

e" X2 (1). - T')-' d1.

= X + etr, 2

and then (25) implies X' erT2 QS = X' e1T =

for all t e R.

Hence QS must be the identity on W.

We now turn to the problem of constructing a y-spectral triple for a product L(A) = L2(i.)L,(i.) of matrix polynomials given a y-spectral triple for each factor. As shown in §4.3 there is a unique subspace 2 such that

Further results

87

(X; , T;) = (X+I2, T+Iu) is a 7-spectral pair for L1(;.). Let Y+ e 2(C°, Q) be the unique operator such that (X+, Tl , Y;) is a y-spectral triple of L,(i.). We claim that the operator S from Lemma 4.15 satisfies the equations X+S =

1

I

tai

L 1(i.)XI (Ii. - Tz)-' di.

(28)

and

T+S=ST2 +Y;X2,

(29)

Of course (28) follows from (27) by letting t = 0. Also (29) is equivalent to

X+ e'r' T+S = X+ e'r. ST2 + X; eITi Yi Xi

However, in view of (27) and the fact

due to the injectivity of col(X+ that

X; e'T1, yl =

1

I

(30)

e'z Xi(11.

- Ti) ' It di. = 1 J

2ni ,

2iri f7

e`'` Li 1().) d%,

we see that the right-hand side of (30) is equal to

J2ni I

etz

L, 1(1)X2 (Ii - TT)-1T2 d ; . +

v 1

2ni

.,

1

2 ti

2L .f

e'k L, '(i.)XZ di.

e''`L- 1(;)Xi((1A-Tz)-1T2 +1}di. e'" Li'(i.)).X2 (11. - T2+)-' d).

= X+ e'r. T+S, which is the left-hand side of (30). This proves (30) and hence (29).

Lemma 4.16 Suppose that S e Y(9R2 , 9R+) and Q e.'(91t+, 9R;) such that QS = I (the identity operator on 02+), and let 9711 = ker Q. Then the operator M e .'(9)l+ x 9R2 , In +) defined by MI

c1)=c,

+Sc2,

c,eSl7ti,c2e9R2,

c2

is an isomorphism with inverse M-' =

(s), where P = I - SQ e.(9n+) is

a projector with im P = 9n; and ker P = im S. The proof is left as an exercise.

In the following theorem, L,().) and L2(i.) are p x p matrix polynomials and )' is a simple, closed contour not intersecting sp(L1) or sp(L2).

Boundary value problems for elliptic systems

88

Theorem 4.17 Let (Xj , T;, Yj) be a y-spectral triple of L,{1) and denote the base space by W (j = 1, 2). Then the product L(1.) = L2(i.)LI(A) has the following ,,-spectral triple:

X+=[X- A],

T,

7X2

fit

Y+=( ).

J,

(31)

where

A=1 J Ll '(i.)X2(I1-T2)-' d2. 27ri

1f

.,

B=

27ri

Y;}di.

,

Proof We begin by letting (X+, T+, Y+) be any y-spectral triple of L(1), not necessarily of the form (31), with base space denoted 972+. Without loss of view of Lemma 4.16, the operator generality (X', T,) = (X+I e, Tl e) In(C2) M e 2(972; x 972; , 9)2+) defined by

Mc,= c, + Sc, is an isomorphism.

We claim that 111-iT+M _

X+M=[Xi A],

' zl

(O

-CT+

M- IY+_

X+/

Y+

( Y2+ l B / (31' )

(This is sufficient to prove the theorem since an admissible triple that is similar to a y-spectral triple of LO.) is itself a 7-spectral triple.) By the definition of one verifies easily that, for all c, a 9)ti , c2 E 9)22, X+MCl

M X+cl + X+Scz = [X+1

X+S](c'), cz

and by (29)

(

\

T+MI C" 1 = T+c, + TScz =

M(

'

Y'+

2

In view of (28), the first two equations in (31') hold.

Further, note that the image of Y+ - S); is contained in M'. Indeed, by Lemma 4.13, we have Q Y+ = Y- so that

Y+-SY2 =(1 -SQ)Y+=Pf+. Since the image of P is equal to 9)l (Lemma 4.16), the image of Y+ - SY2 is a subspace of T1* and then

M Y+-SY2 Y;

=Y+-SY2 +SY2=Y+.

Further results

89

Hence the last equation in (31') holds provided we show that Y+ - SYZ = B. But consider the fact that

X. e".(., - SYZ) = X. e` Y+ - X+ err. SYZ, so that by virtue of (27) we obtain

X+eIT.(Y+-SYZ)=2I 1ri

=

=

1

I

f e'"LikLZ'(2)-XZ (IA-T2)''Y2}d). I e'.1 X'(li.-Ti)Yi

2ni J,

x 1, L2 X i eIT i B

X2 (1). - T2+

Yi i d).

= X+ e'T * B.

Since col(X+T+);=o is injective, it follows that Y+ - SYZ = B. Remark Theorem 4.17 generalizes the result of Theorem 3.2 for if we assume

that L,().) and L2(i.) are monic and consider standard triples for L,().) and L2(;.) (i.e. let 7 be a contour having all of sp(L1) and sp(L2) in its interior), it is easily seen that A = 0 and B = 0.

It is convenient for some purposes to state Theorem 4.17 in a modified

form. Recall that we have let 2 denote the unique subspace such that (X+I,L, T+I,,) is a y-spectral pair of L1O.).

Theorem 4.17' Let (X+, T+) be a 7-spectra! pair of the product L().) _ L2().)L1().). Also let (Xj, Tj) be a ;-spectral pair of Lt().) (j = 1, 2). Then there exists a unique isomorphism M e 2(TIi x Tl2, M*) such that X+M=[Xi A], T+M=M T+` Y+ X+2 1

0

TZ

where A is defined in the statement of Theorem 4.17 and Y' a Y(C , W) is

the unique operator such that (X; , T', Yl) is a i-spectral triple of L,Q). Further, if (X T) _ (X+I2, T+I a), then M has the form M(C2 C'

J

= c, + Sc2,

where S e 2'(D12, 901 `) is the unique operator satisfying the equations (28), (29).

The existence of M follows from the proof of Theorem 4.17 and uniqueness

is a consequence of the injectivity of col(X+T+);=o. The fact that there is only one operator S that satisfies (28), (29) also follows from the injectivity of col(X+T+);=o for if both S and S' satisfy (28), (29) then X+(S - S') = 0 and Tr - (S - S') = (S - S') TZ which implies

X+T+(S-S')=0 whence S - S' = 0.

j=0, 1,...

90

Boundary value problems for elliptic systems

Remark If we choose the first two operators in (31) as the y-spectral pair (X+, T+) then M = I. It follows in this case that the operators S and Q in Lemmas 4.12 and 4.15 are =C2,

SC2 =

CIEW1 t+,C2 E02+

(C02

We now introduce rectangular matrix polynomials (these occur in boundary problems for elliptic systems): m

i = 1, 2

B,(A) _ Y B( )A1, i=o

of degree m, where the coefficients Bj ' are r, x p matrices, and then let B(A)

=

B1().) B2(A)L1(A)

=m

Pit

1

which is a matrix polynomial of degree 5 m = max(ml, m2 + 11) where the coefficients fl are (r, + r2) x p matrices. (Ti )t, i = 1, 2, and A = Y7_o A,X+T+. Corollary 4.18 Let A, _ E'°_-o A' Z where M denotes the isomorphism in Theorem 4.17' Then A-M = 0 AZ , and

Z = 2'i , B1(A)Li (A) X,' (1i. - Ti)' -dA Proo f The isomorphism M: tO1i x U12 - Wl+ is defined by Mcl

()

c1 + Sc2. Note that A = B(d/dt)X+ e`T' I,=o, so that if c, e M then Act = B(d/dt)Xi e`T' c11,=o

C2

Xi e,T, B,(d/dt) 1,=o c1 B2(d/dt)L1(d/dt)

= EE

0 1) C°1`

where we used the fact that L1(d/dt)X; err' = 0 since (X', T,+) is a y-spectral pair of L,(A). Next we use (27) to obtain A-S = B(d/dt)X+ e,r. SI'=o I

B1(d/dt)

( B2(d/dt)L1(d/dt)) 2ai j1

e,z LT '(;.)X+ (IA - Ti)-1 dAl,=o

Z 2ai r

2()

B,, (A)

=( AZ). The proof is complete.

2

+-1 (1). - TZ)

d).

Further results

91

45 Transformations of products Let L(A) and L1(A) be p x p matrix polynomials such that L(A)LI '(A) has an analytic continuation inside y. Let (X+, T+) be a y-spectral pair of L(A). As shown in §4.3 the subspace 2 = {7; 7 e SA2+, L,(d/dt)X+ etT+ 7 = 0}

is the unique T+-invariant subspace of '.172+ such that (X+IB, T+Ie) is a y-spectral pair of L1(A). Let cp(A) = (aA + b)/(cA + d) be a Mi bius transformation as in §2.4 with

-d/c outside y. The transformed matrix polynomial r

f(A) = Y A/d2 - b)1(-cA + a)'-1= (-cA + a)'L((p-'(A)) 1=0

has the y-spectral pair (X+, T+) which is defined in Theorem 2.21. Ordinarily we choose I to be the degree of L(A), i.e. the minimal integer such that A1= 0

for j > I, but it will be important for the proof of the lemma below to be able to increase 1 arbitrarily by adding on zero coefficients. This has the effect

of multiplying L(A) by powers of -cA + a, but since a/c lies outside y' this does not change the spectrum of L(A) inside y', nor does it affect the y'-spectral pairs (see Corollary 4.9).

The following lemma implies that the subspace 2 is unchanged by the transformation -, that is, P- = Q. Lemma 4.19 (X+le, T+le) is a y-spectral pair of L1(A).

Proof Since a/c lies outside y, the number of zeros of det L1(A) inside y is equal to the number of zeros of det L1(A) inside y'. Also

L(A)Li '(A) = (-cd +

% p - ' ( ). ) )

is analytic inside p so there is a unique T+-invariant subspace D such that (X+Ik, T+I0) is a 7-spectral pair of L1(7.). To prove the lemma it suffices to show that L,(d/dt)X'+ e`T*,, =- 0

for all 7 e 2

(32)

This would imply that 2 e i, and since dim D = dim 2 = r1 (the number of zeros of det L1(A) inside y, or det L1(A) inside then 2 _ 2 must hold. Without loss of generality, by adding on zero coefficients to either L(A) or L,(A), we may assume that the same "degree" is used in the definitions of L(i.) and L1(A), that is, I = 11. We will use formulas (20), (21) from Theorem 2.21. As in the proof of that

theorem we assume without loss of generality that ad - be = 1. Let the coefficients of L,(1.), L,().) be denoted Al", Ai", respectively, j = 0,..., I. By 2.4(20) we have

[2'

v,

Boundary value problems for elliptic systems

92

and by making use of 2.4(21) r

[Aor...A(1'r]-col(X+T+)jo0

i AjrrX+T+ = 1=0

_ [A(".. A'')] col(X+T+)j=o(cT+ + dl)

_

dl) j=0

Thus

L1(d/dt)X+ eti. q = i Ak"X+T'+7, i=o

(33)

(cT+ + dl)-' e`r q (for a fixed r e R). Now let q e 2. Since 2 is invariant under T+ it is also invariant under cT+ + dl. But -d/c ties outside y so cT+ + dl is invertible and 2 is therefore invariant under the inverse where

operator (cT+ + dl)-'. Since T+ _ (aT+ + bl)(cT+ + dI)-' then t,(2) C-21. Hence ii a i2 and, in view of (33), it follows that (32) holds.

We now consider the effect of the transformation - on a product of matrix polynomials, LO.) = L2(A)L1(A), and on the operators Q and S associated with this product as in §4.4. Let l,, 12 and I denote the degrees of L1(i.), L2(1) and L(i.). Then I < 1, + l2 and we have (-c). + a)"+'2-'L(1.) = L2(A)L,(A). By adding zero coefficients

to L(1.) (see the remarks preceding Lemma 4.19) we may assume that I = 1, + 12 and then L(i.) = L2(i,)LL1(i.).

In §4.4 the operators Q, S enabled us to construct y-spectral triples of a product in terms of y-spectral triples of each factor. The question we now consider is this: what is the relationship between the operators Q, S defined with respect to the product L2(i.)L1(1.) and the operators Q, S of the original product L2(i.)L1(i.)? This question is answered by Proposition 4.22 below. As usual, (X+, T+) is a y-spectral pair of L(..) with base space 10+, and

(X+, t.) denotes its transformation under (p as in Theorem 2.21. Also (Xi , T,) is a y-spectral pair of L.(i.) (i = 1, 2) with base space 'M1, with (X; , T,) the corresponding transformation under (p. Without loss of generality we may assume that (X i , Yl) = (X+le, T+where 2 is the subspace (24). In view of Lemma 4.19, (X; , T,) _ (X+li, T+li) for the same

subspace 2. Also we let Yi be the unique operator such that (X', T; , Y') is a y-spectral triple of L1(A); then (9+, t+, R+), its transformation under (p, is a y-spectral triple of L1(i.). Lemma 4.20 Under the hypotheses and notation as stated above, we have

T+S=STT +(-cTi +al)Yt-XZ(cTT +dl)-' Proof By Theorem 4.17'

X+ =[Xi

A]M-1,

T+=M

T+ U

Y+ '7+X2 2

(34)

Further results

93

where M(xY) = x + Sy. Then

T+ _ (a T+ + bl)(cT+ + dl)

= M(aT, +bl aY, - X,+ )jcT, +dl cY1 - X,+ 1-' M_1 0 aTz +b111 0 cT2 +dI)

and, using the fact that for an u per triangular block matrix of the form 0

the inverse is given by

B

/ T1

T+M=M1

0

I

-A-'CB-1), it follows that B1

(-c1; +al)YtXZ( c T2 +dI)-' ) Ti

0

Now (34) is an immediate consequence of (35) since M

G°"

(35)

= Sy.

Lemma 4.21 We have

X+S = (ad - bc)-''X+(cT+ + dl)" -'S(cT2 + dl)

(36)

Proof In view of (28) in §4.4 we have

1

+

L(1.)Z(11.-TZ)-'d;.

2m 1

2ni

1

(-ci. + a)-"Li'((p-'(A))Xi (Ii. - (p(Tf ))-' dA

9

and, with the substitution A = (p(w) = (aw + b)/(cw + d), it follows that

X+S =

1

27ri

r (-ca(w) + a)-'1L1 '(w)Xi (1p(w) - q (Tf ))-' ad - be

(cw + d)Z

r

Since -ccp(w) + a = (ad - bc)/(cw + d) and

10W) - OTz) =

ad - be

cw+d

+ (Iw - T+z)(CTz +dl)- , 1

we obtain

X+S = (ad - bc)-'' 27ri

x

f (cw + d)'-Lj'(w)XZ (1w - T2+)-' dw (cTZ + dl)

_ (ad - bc)-''X+(cT+ + dl)" -'S(cT2 + dl).

dw.

Boundary value problems for elliptic systems

94

Proposition 4.22 We have also

S = (ad - bc)-"(cT+ + dl)" -'S(cT2 + dl)

(37)

Q = (ad - bc)''(cT2 + dl)-''Q

(38)

and

rI

Z

0

(ad - bc)-''(cT2 + dl)"

MI

(39) I

where I is the identity operator on 9Ri and Z is the operator from 9Jz to 9J1; given by

Z = c(ad - bc)-"

(c T+ + dl )j Y1 X 2(c T2 + dl )k+'

(40)

j+k=11 -2

Proof Let S' denote the right-hand side of (37). In view of Lemma 4.21, we have X ,.S = X+S'', and hence

X+(S-S'')=0.

(41)

Also, by Lemma 4.20, we have

T+S=ST2 +(cTi +al)Yi XZ(cT2 +dl)-' ; _ (ad - bc) - "(cT+ + dl)" -2 Yi and Then, since XZ = X 2, _CT' + al = (ad - bc)(cT, + dl) ', it follows that T+S = STZ + (ad - bc)''(cTt + dl)-('' -1YlX+(cT2 + dl)-l.

Upon multiplying this equation by (ad - bc)-"(cT+ + dl )" -' on the left and cT2 + dl on the right, we obtain T+S' = S'T2 + Y; X2.

(42)

However, due to (28) and the definition of S', we also have T+S =STZ + Y; XZ and subtraction of (42) from this equation yields (43)

TT

Now (41) and (43) imply that

X+Tf+(S-S')=0

forallj=0,1,....

Hence S - S'' = 0 since col(X+T+);=o is injective. Thus S'' = S and (37) holds. Next we prove (39). By the corollary to Lemma 4.23 (see below), it follows

from (37) that

S = (ad - bc)-"S(c/T2\+ dl)" + Z,

whereZisgivenby(40).Then,sinceMl X 1=x+Syandif (X) =x+Sy y

for all x e St and y e 9712, it follows that (39) holds.

y

Further results

95

To prove (38), recall from Lemma 4.16 that M` = (s). Thus Q = [01 ] M -1, where I is the identity operator on 9122 ; similarly, Then

[01] M - '.

Q = [01]M[01]

1

0

(ad - hc)"(cT2 + dl)-`'

M-

= (ad - bc)"(cT2 + dl)-''Q. The asterisk * denotes the operator -Z(ad - bc)11(cT2 + W)-11, and is of no significance in the derivation of the formula (38). This completes the proof of the theorem.

Remark If the numbers a, b, c and d are multiplied by k # 0 then the Mobius transformation (p is unchanged but L1(;.) is replaced by k''L1(i.) and then

S is replaced by k-''S and Q is replaced by k''Q. This is the reason for the

factors ad - be in the formulas (37) and (38). Note that by choosing k = (ad - be)-', we may assume without loss of generality that ad - be = 1 in all formulas in this section. Remark If the formulas (37) and (38) are inverted, we obtain the following:

S = (-cT+ + al )'' -'S(-cT2 + al), Q = (-cT2 + aI)-''Q, and the analogues of (34) and (36) are

T+S=STT +(cTj +dI)Yl XZ(-cT2 +al)-' and

X+S = X+(-cT+ + aI)''-'S(-cT2 + al). Lemma 4.23 T'+S = S(TZ)m +

j+k=m-1

(Ti)'Yi.X2(T+)k

(44)

Proof By (29) of §4.4 we have

T+S = ST2 + Y1 X2,

so that the lemma holds if m = 1, and then we may proceed by induction on m.

Corollary For any real numbers c and d, (cT+ + dl )'S = S(cT2 + dl)"' + c

Y j+k=m-1

(cT1 + dl )jY1 X 2(cT2 + d1)k

96

Boundary value problems for elliptic systems

Proof The formula (44) holds if T. + al is substituted for T+ (since (29) remains true). Now if c:0, simply let x = d/c. In §16.2 we will need a generalized version of Corollary 4.18. Proposition 4.24 In addition to the hypotheses and notation of Corollary 4.18,

let ,(i.) = W. + b)/(cA + d) be a M6bius transformation with - d/c in the exterior of y. Let B;(1.) _ m=o B f'1.j denote the transformation of B;(i.), i = 1, 2. Also let

LJ

B2(i)L1(.)

j=o

J

Without loss of generality, ad - be = 1, and then the following formula holds:

Al

A-M

*

(0

+dI)

where

e = X fijX+T +, j=o

j=0

and (X+, t,) and (21+, T+), i = 1, 2, are the transformations of (X+, T+) and (X; , Ti ), respectively. The asterisk * denotes an operator from SD22 to C" for which an explicit formula is given below.

Remark B(i.) itself is not the transformation of B(1.), unless m,- = m2 + 11.

Also, the transformation of L2())L,(1.) is not equal to L2(i.)L,(i.) unless 1=11 + 12; see the discussion preceding Lemma 4.19.

Proof We first apply_ Proposition 4.18 to the product L2(A)L'1(A) and the matrix polynomials B,().), B2(i.) and Be.) to obtain e,

z A2,

o

where

Z=2ni

-TZ)-1di.. y

Also, by Proposition 4.22 we have 1

W

M(0 (cT2 + dl)'' where (c T,+ + dl )JY; X +(cT2 + dl )k+ 1

W=c j+k=1, -2

Further results Therefore

A-M = AM.(M-'M)-' -' Z 1 w AZ)(0 (cTT + dl)'') _ Al Z 1 - W(cT2 + dl)-', (0 A,)(0 (cTT + dl)-''

Al

0

-

(A' 0

A2 (cTT

+d1)-'')

where the asterisk * denotes the operator (-A1 W + Z)(cTT + dl)-1i.

97

Part II Manifolds and Vector Bundles

99

5 Manifolds and vector bundles

5.1 Background and notation We begin with a brief review of differential calculus in finite dimensional spaces. Let E and F be finite dimensional spaces over f8 with norms denoted by 11. (All norms are equivalent on finite dimensional vector spaces.) A function f: U - F where U is an open subset of E is differentiable at x E U if there is an element Df(x) E L(E, F) such that

If(x + h) - f(x) - DI(x)hl = o(IhI), h-0 Here L(E, F) is the space of linear transformations from E to F, which we endow with the norm IITII = sup,.,,, ITxI. The notation o(t) denotes a real-valued function of a real variable such that lim,.o o(t)/t = 0. We denote by C'(U, F) the set of continuously differentiable functions from U to F, that is, the set of functions f: U - F which are differentiable at every point and for which U -3 x r-. Df(x) e L(E, F) is continuous. We will often need to use the chain rule which states that if f: U -+ V is

differentiable at xo (U open in E, V open in F) and if g: V -. G is differentiable at f(xo), then g o f is differentiable at xo and

D(g'J)(xo) = Dg(f(xo)) ° Df(xo) The formula shows that if f and g are of class C' then so is g o f.

(1)

Let f: U c E - F be of class C', that is, the map

Df: U - L(E, F) is continuous. Since L(E, F) is itself a vector space we can consider the second

derivative DZf = D(Df) which, if it exists, is a map U - L(E, L(E, F)). The space L(E, L(E, F)) can be identified with LZ(E, F), the set of bilinear maps

E x E - F, so we have DZf: U - LZ(E, F),

and we say that the value of DZf (x) at (h, v) E E x E is DZ f(x)(h, v) Higher-order derivatives are defined inductively. Let E(' = E x xE and let L'(E, F) denote the space of r-multilinear maps from E to F, that is, the set of maps E°" - F which are linear in each argument separately. We 101

Boundary value problems for elliptic systems

102

endow the vector space L'(E, F) with the norm 11 T11 = sup,.,,,,,, I T(x1, ... , x,)I

Note that L(E, L'-'(E, F)) can be identified with L'(E, F); the identification

is given by the equation T(x...... x,) = T(x,)(x2, ... , x,) and is normpreserving. The rth derivative D'f is defined as D(D'''f) and is a map of U into L(E, L'-'(E, F)), so

D'f:U-'L'(E,F) A function f: U -' F is said to be of class C' if all derivatives Dkf: U Lk(E, F), I < k <, r, are defined and continuous. We denote the set of all such maps by C'(U, F). Note that if F = 68 we write L'(E) = L'(E, R) and C'(U) = C'(U, R). If f is differentiable at x e U we say that Df(x) v is the directional derivative

of f at x in the direction v e E, and it follows from the chain rule that the directional derivative is given by

d f(x + tv)I,=,

(2)

By repeated application of this result we obtain the values of D'f(x),

D'f(x).(v,,

v,) =

d

dt...-d f(x + t,v, + ... +

(3)

r

a formula which is often convenient for calculation of higher-order derivatives.

Example

(i) If f e L(E, F), a linear map E -+ F, then Df(x) = f for all x e E. Since Df is constant (independent of x), it follows that D2f = 0. (ii) If f e L2(E, F), a bilinear map E x E - F, then I f(xi + h1, x2 + h2) - f(xl, x2) - [f(h1, x2) + f(x1, h2)]1= If(h1, h2)I, and the right-hand side is o((h1, h2)I), where I(h1, h2.)I = (0112 +

('212)112.

It follows that the derivative Df(x1, x2): E x E -+ F is given by Df(x1, x2)-(h1, h2) = f(h1, x2) + f(x1, h2)

Differentiating once more we obtain D2f(xl, x2)-((h1, h2), (k1, k2)) = f(hl, k2) + f(k1,h2) Hence the value of D2f is constant (independent of x = (x1, x2)), and therefore D3f = 0. (iii) If f e L'(E, F), a multilinear map E(") -+ F, then

Df(x1...... ,)-(h1,...,h,)=f(h1,x2,...,x,)+f(x1,h2,...,x,)+.. + f(xj,...,x.-1,h,) Proceeding inductively, it is clear that f is C'° and Dkf = 0 when k > r.

Manifolds and vector bundles

103

Proposition 5.1 If f: U -, V and g: V -+ G are of class C', then g - f is also class Cr.

Proof The map gyp: L(F, G) x L(E, F) -+ L(E, G) defined by (A, B) " A - B is bilinear, whence C. By the chain rule (1), D(g - f) is the composition of two maps: G) x L(E, F) -2+ L(E, G)

x i-- (Dg(.f(x)) Df(x)) i- Dg(f(x)) ° Df(x),

where the first map is C'-' since f and g are C', and the second map is C°°. Thus D(g - f) is C'-', hence g - f is C'. Let L(E) = L(E, E) and let GL(E) denote the set of invertible maps (linear

isomorphisms) in L(E). Note that GL(E) is an open set in L(E) for by choosing a basis in E we can assume that E = a", and an n x n matrix is invertible if and only if its determinant does not vanish. (Also see Exercise 5.)

Proposition 5.2 Let I: GL(E) - GL(E) be defined by A F-+ A'. Then the map .4 is of class C°- and for the first derivative DJ(A): L(E) -+ L(E) at A e GL(E), we have

DJ(A)B= -A-'-B-A-' Proof If we knew already that the map .Q

is differentiable then, by differentiating the equation A - A -' = id, the formula for the derivative D,i would follow easily. In fact it is clear that .4 is Cm because we can assume E = 68" and then use the formula A -' = A`°/det A where A`° is the cofactor matrix of the n x n matrix A. We will give another proof, however, which has the advantage that it holds when E is any Banach space. Let A e GL(E). Observe that

(A+H)-' -A-' +A-'-H-A-' (A+H)-'-[A-(A+H)+(A+H)-A-'-H]-A-' =(A+HY',H-A-'°H-A-' and the last expression

is o(IIHII). Hence .1 is differentiable and DJ(A)H = -A-'HA-'. To prove that I is C°° we proceed by induction.

The derivative DJ can be expressed as a composite of two maps

DJ: GL(E) - GL(E) x GL(E) -+ L(L(E), L(E))

A,-4 (A-', A-') F-- -A-'(,)A-' The second map, (p, is the restriction of the bilinear map L(E) x L(E) -+ L(L(E), L(E)) given by (A, B) r and is hence C. If we assume as the inductive step that I is C' then DJ is the composition of a C' map and a C'° map. Thus DJ is C', hence I is C'+' Let f: U -+ F where U is an open set in E, x E2 and E E2 are any finite

Boundary value problems for elliptic systems

104

dimensional vector spaces. Let x = (x1, x2) E U. The derivatives of the mappings y, '--' f(yl, x2) and y2 -+ f(x1, v2), if they exist, are called partial derivatives of f at the point x and are denoted by D, f(x) e L(EI, F) and D2 f (x) e L(E2, F), respectively. Then we have the following proposition.

Proposition 5.3 Let U c E, x E2 be open and f: U then the partial derivatives exist and we have

F.1f f is differentiable,

Df(x).(h1, h2) = D1f(x)-hI + Moreover, f is of class C' if and only if D; f: U --+ L(E;, F), i = 1, 2, i = 1, 2, both exist and are of class C'- '

Let L'(E, F) denote the space of symmetric r-multilinear maps, that is, maps in L'(E, F) whose value at (v1..... v,) is not changed by a permutation

of v1,...,v,. Proposition 5.4 If f: U (-- E - F is C' then D'f(x) e L,(E, F), i.e. D'f(x) is symmetric.

The notation for partial derivatives of a function defined on open sets in 18" is as follows. If f: U e 08" -+ R' is a vector-valued function we write out

f in component form

f(xl,

,xn)=[f1(xl,...,xn),...,fin(xl, .... xn)]T

where each f : U - R is a real-valued function. The superscript T indicates transpose of a matrix, that is, we regard the values of f as column vectors in 18m. The total derivative Df(x): RI -+ 18" can be regarded as an m x n matrix

Df(x)=I of

axj

mxn

where the rows are indicated by the index i = 1, ... , m and the columns by j = 1, . . . , n and a fl/axj is the derivative of f, with respect to x j, keeping the other variables x1..... x_1, xj+ 1i ... , x" fixed. In view of (3) the components of D'f(x) in terms of the standard basis e ,..., e" in R" are

D'f(e;...... el,) =

a

axj,

-..

Of

(4)

ax,,

Proposition 5.3 implies that f is of class C if and only if the partial derivatives (4) are defined and continuous for all r g s. By Proposition 5.4 we can write (4) in the form

a,,-..a-1°f

(aj = alaxj)

where a= (al, ... , an) is a multi-index, i.e. an n-tuple of non-negative integers. For short we write as = a;" . . a." and let Pal _

aj = r denote the

Manifolds and vector bundles

105

order of differentiation. By (3) we have

(r)318

Drf(x) (h, ... , h) = IQI =r

where

(r) = r!/a!,

a! = a l!

and

. a"!

(5)

a

h' = h;'

.. hl."

where

h=

Of

(h...... h")eR". Taylor's formula

Let [x, y] denote the line segment {x + t(y - x); 0 < t < 1) between two points x, y e E. If f e C1(U, F) where U is a neighbourhood of the line segment [x, y) then we have d/dt f (x + t(y - x)) = D f (x + t(y - x)) (y - x), and integrating this equation with respect to t we have

f(Y) - f(x) = f, Df(x + t)y - x))' (Y - x) dt

(6)

0

In the statement of Taylor's formula below we use the notation h'"' = (h, ... , h), where h is repeated k times. The notation therefore, the value Dkf(x)(h, ... , h).

means,

Theorem 5.5 Let x, h e E and let f e CN(U, F) where U is a neighbourhood of the line segment [x, x + h]. Then

I

f(x+h)= 1

(N-1)! where

Rv(x,h) =

DN - 1f(x) . h(N - 1) + RN(x, h)

(1 - ON -I DNf(x + th).h(NI dt fo,

(N- I)!

Proof For N = I this is just (6). The theorem can be proved by induction on N. If the formula holds for some N then it holds for N + I by integration by parts on the integral RN(x, h) since dt

DN+1 f(Y +

DNf(x +

which follows from (2).

Remark 5.6 If f: U c R" - R' then, due to (5), Taylor's formula takes the form

f(x + h) = Y

I

III
a°f(x)h° + RN(x, h)

Boundary value problems for elliptic systems

106

where 1

(1 -

N

RN(x, h) = lal=N

ON- ` 1 a=f (x + th)h° dt 71

o

Homogeneous polynomial maps

If T e L'(E, F) then the map p: E -+ F defined by p(x) = 71x,..., x) is called a homogeneous polynomial map of degree r from E to F. The set of all such

maps is denoted P(')(E, F). We also define P(0)(E, F) = F, the space of constant polynomials. Each permutation e e S, defines a map a: L'(E, F) -+ L'(E, F) by (aT)(V1, ... , v,) = T(va(1), ... , va(r))

Thus by definition T e Lr(E, F) (i.e. T is a symmetric r-multilinear map) if aT = T for all or e S,. Define the linear map Sym: L'(E, F) - L;(E, F) by

SymT= 1 Y aT 1.106s, We have Sym T = T if and only if T E L"(E, F), and the operator Sym is a projector, i.e. Sym o Sym = Sym.

Proposition 5.7 Let p e P(')(E, F), i.e. p(x) = T(x, ... , x) for some map T e L'(E, F). Then for all x e E we have 1r'

D'p(x) = Sym T

(7)

.

Hence for any p e P('i(E, F) we have p(x) = D'p(0) (x, ... , x) jr!. The map

L,'(E, F) - P(')(E, F) defined by Ti- p is an isomorphism; its inverse is p

D'p(0)lr!.

Proof By virtue of (3), we have d

dtr

d

P(x+Y-

(8)

1

Since p(v) = T(v,... , v), where T is multilinear, the expression p(x + E tivi) is polynomial in the variables t,, ... , t,. It is clear that the only terms that remain after substitution in (8) are those that contain the product t1 - t,. In particular, it follows that (8) is independent of x. So let x = 0. Then we have p(

t,vi) = RE tiv1, ... , I t1v1) = Y tip ... ti,T(vi, . , Vi.)

by multilinearity

and the terms in this sum that contain the product t,

t, correspond to

Manifolds and vector bundles

107

permutations (i1, ... , i,) e S,. Hence, substituting in (8), we get D'p(x)(v1, ... , v,) = Y T(v;...... v,,) where the sum is taken over all permutations a = (i1, . . . , i,) e S,. This proves the first statement (7). The second statement also follows immediately. To complete the proof observe that if p(x) = T(x, . . . , x) then we can replace T

by Sym T, so without loss of generality T is symmetric, in which case (7) gives D'p(x)/r! = T so we can recover T E ,(E, F) from p. Inverse function theorem

A map f: U - V, where U is open in E and V is open in F, is said to be a C' diffeomorphism (r > 1) if it is of class C', is bijective, and has an inverse g: V -+ U which is also of class C'. By virtue of the chain rule it is clear that if f is a diffeomorphism then the linear map Df(x): E -+ F must be invertible

at each x e U; in fact, by differentiating the equations f o f -' = id and f -' of = id, we infer from (1) that D f-1(y) = [Df(f - 1(y))] - 1 (9) For the converse we have the Inverse Function Theorem, the proof of which can be found in any textbook on real analysis, see [Ca] or [La 1].

Theorem 5.8 Let U be an open subset of E and let f: U

F be of class Cr.

Assume that for some point xo a U, the derivative Df(xo): E - F is an invertible map. Then f is a Cr diffeomorphism of some neighbourhood of xo onto some neighbourhood of f(xo).

The main part of the proof is to show that f is a homeomorphism from some neighbourhood U of xa onto some neighbourhood V of f(xo). Then one shows that (9) holds in a smaller neighbourhood V' e V of f(x0) and

f -1: V' - U' c U is the inverse in that neighbourhood. Since (9) can be written in the form

Df-' =,poDfo f-'

(10)

it follows from Proposition 5.2 that Df -' is continuous, whence f -' is C'. In that way one shows that f: U' -+ V' is a C' diffeomorphism. By induction using (10) one then shows that f -1: V' - U' is of class C'. Exterior product of a vector space

The exterior product of a vector space is needed later when we define differential forms on manifolds. For the convenience of the reader we write down the definitions and basic properties of such products. For the proofs, see [La 2]. Let E., i = 1, ... , r and F be finite-dimensional vector spaces over R. A

map f: E, x

x E, - F is said to be multilinear if it is linear in each

variable separately. We denote the set of such maps by L(E,..... E,; F). If

Boundary value problems for elliptic systems

108

E. = E for all i we write E" = E x x E and denote as before the set of r-multilinear maps E" - F by L'(E; F). An r-multilinear map A: Et' -+ F is said to be alternating if A(v,, ... , v,) = 0 when v; = v;, i 0j. This condition can be stated in another way: the sign of A(v , ... , v,) changes if any two of its arguments are switched. The set of alternating maps Et' -, F is denoted L' (E, F). For each positive integer r there exists a vector space, n'E, and an r-multilinear, alternating map, E" -+ n'E, denoted by A ... A V,,

which is universal with respect to r-multilinear alternating maps on E. By "universal" we mean that if g: E(') - G is an r-multilinear alternating map, then there exists a unique linear map gs: Ar(E) -+ G such that the following diagram is commutative:

E''- A'(E) 19.

G

The universal property characterizes NE up to a unique isomorphism, and it is called the rth exterior product of E. An element of /VE of the form V, A A v, is said to be decomposable. Every member of NE is a linear combination of such decomposable elements. Note that A'E = 0 for r> n, where n = dim E. Now let 1 r n. It is a consequence of the universal property that if e,, ... , e" is a basis of E then the elements

it<...
e,, A ... A ej,,

form a basis of NE. Thus, the dimension of NE is ('i). For r = 0 we let

nE=R.

For each pair of positive integers r, s there exists a unique product

(bilinear map)

NE x NE - /V`E

such that if v1, ... , v w,, ... , ws E E then A

(V1 A

V,AW, A

A

W.,)

Ws

This product is associative. The exterior algebra, A(E), is the direct sum 0 NE, that is, A(E) = R ® NE ® A2E ® ... ® NE, with multiplication defined by

e

1

r=0

z) A (

Ys) 0

_

1

k=0

Y ar A ins)

(1 1)

r+s=k

It is easy to verify that A(E) is an algebra over R in the sense that it is a vector space over R and the multiplication A is a bilinear map. The

Manifolds and vector bundles

109

bilinearity of A means that for all c e R and a, /3 and w in A(E), (ca) A Q= I A (c/3) = c(a A /3)

(a+1)Aw=aAw+/3Aw wA(a+ fl)=wAa+WA/3 Further, the multiplication is associative, that is, a A (/3 A (0) = (a A /3) A w for all a, /3 and to in A(E), so (E) is an associative algebra over R. Also,

note that if a, e n'(E) and P. a '(E) then

P. A a,=(-

1),Sa,

Aft

(12)

It follows from (12) that if /3 is a sum of terms of even degree then ocommutes with any /3 a A(E).

Remark As a vector space over/l, IA(E) has dimension +(n)

+1

2J+...+(i) =2".

1

If f e L(E, F) is a linear map, then, due to the universal property of exterior

products, there is a unique linear map A'(f) e L(n'E, n'F) for r = 1, 2, ... , that maps

V1 A...Av,t-'f(vl)A...Af(v,)

Also, we let A°(f) a L(R, R) be the identity. Note that / +S(f)(a A /3) _ n'(f) A 8(f)(0) for all a e n'E and /3 e ASE. Each map A'(f) is linear so there is an induced linear map A(f ): A(E)- A(F) on the direct sums, and it is evident that A(f) is a homomorphism of algebras, that is,

A(f)(a A /3) = A(f)(a) A A(f)(fl)

(13)

for all a and /3 e AE, where the multiplication A is defined by (11).

If E and F are vector spaces over I8, there is a natural isomorphism of algebras over R

NE (B F) = A(E) O+ A(F) R

such that the vector space Ak(E p+ F) is isomorphic to ©;+;=k n'E ® I VF for all k = 0, 1, 2,... Later on we will be working with the exterior product A'(E*) of the dual space E*. Sometimes it is convenient to identify it with L.'(E, R), the space of alternating r-multilinear forms on E. Theorem 5.9 There is an isomorphism A'(E*) L'(E, R) which maps ' A . A S' e n'(E*) to the r form (v1, . . . , v,) -* det[ j :1.

Sketch of proof If S', ... , S' a E* and v1..... v, e E then the value is an alternating, multilinear map as a function of both

110

Boundary value problems for elliptic systems

(', ... , Sr) and (v...... v,), hence defines an alternating r-multilinear map

E*''' - L;(E, R). By the universal property of the exterior product (with G = L;(E, R)) there is a uniquely determined linear map

A'(E*) - La(E, l)

(*)

Choosing a basis e,, ... , e in E and the dual basis i'. ... . J." in E*, it is not hard to verify that the basis vectors i." n A i." Of n'(E*) are mapped to basis vectors in Lo(E, R). Hence the map (*) is an isomorphism.

From now on we identify the element 51 A

A , E A'(E*) with its

image under this isomorphism, i.e. C1 A ... A S,(v1,..., v,) =

(14)

Realization and complexification

A complex vector space E can be viewed as a real vector space together with

a linear map is E - E such that i2 = -1. The underlying real vector space is called the realization of E. Let A: E -' E be a C-linear map. The realization of A, denoted AR, is the same map, but now regarded as an R-linear map. Example The realization of C" is R 'P, where for each v1, v2 e li" we identify v, + iv2 a C" with (v1, v2) a R2D. Given a linear map A: C° -* C", i.e. a p x p matrix with complex entries, we can write A = A, + iA2 where A, and A2 are p x p matrices with real entries. Since

(A1 + iA2)(v, + iv2) = A,v, - A2v2 + iA2v, + iA,v2, the realization of A is the block operator AR =

A,

-A2

(A2

A,

If we take A to be multiplication by i, then iR = (0

01).

Now let E be a vector space over R. Then E ®R C is a complex vector

space in a natural way: c(v (gz) = v ®cz for c e C, and is called the complexification of E. Since C = R a iR, then every element of E ® C can

be written uniquely in the form v, ® 1 + v2 ® i where v v2 a E. As a real vector space (i.e. the realization) we have E ® C ^-- E ® E, where v, ® 1 + v2 ® i is identified with (v1, v2). Multiplication by i is then an C0

-1),

iR-linear map E ® E - E ® E having the block operator form 0 \1 where I is the identity operator on E. An inner product <, > on the real vector space E gives rise to a natural inner product on the direct sum E 0 E by <(v,, v2), (w), K'2)>'= +

Manifolds and vector bundles

III

and to a Hermitian inner product (,) on the complex vector space E ® C by

v,v'eE,c,deC

(v®c,w(D d)-=-cd

There is a relation between the two inner products: if we substitute v = v1 (D 1 + v2 ®i and w = w1 ®1 + w2 ®i into the Hermitian inner product on E ® C, we obtain a bilinear form ((&1, v2), (W1, v'2)) = + + i - i

on the underlying real vector space, E e E. We see that the real part of the Hermitian inner product on E ® C is nothing but the inner product on the real vector space E ® E induced by < , >, that is, Re((v1, v2), (W11 w2)) _ + -

If A: E ® C - E ® C is a C-linear map, we let An denote the same map

A, but viewed as an 68-linear map E ® E - E ® E. We call An the realization of A. Analogous to the matrix example given above, we have the following result.

Lemma 5.10 Let E and F be vector spaces over R. If A: E ®C -' F ®C is a C-linear map then

An =

Al

-A2

A2

Al

for unique R -linear maps A,: E - F, i = 1, 2. Further, given inner products on E and F, let 'A, denote the transpose of A,, and let 'A denote the adjoint of A

with respect to the corresponding Hermitian inner products on E 0 C and F ®C. Then the realization of 'A: Fr ®C - E ®C is (hA)R = ` -:A1

:A2).

A2

A1

Exercises

Prove Lemma 5.10. Also, show that the correspondence A i-, An is a homomorphism from C-linear maps to 11-linear maps, (AB)R = ARBR. Let E be a finite dimensional vector space over R. The dual space, E*, is the set of linear functionals on E, i.e. E* = L(E,11). Recall that if e...... e" is a basis of E. and we define i....... A" a E* by 1.

= bij

then 21, ... , i." is the dual basis of E* corresponding to e1, ... , en. 2.

Let f: E - F be linear. The transpose map f: F* 4 E* is defined by

f(7.) = i.- f, that is, = . Show that if f has matrix [a,j] with respect to bases for E and F, then f has matrix [a,j]T = [aji] with respect to the corresponding dual bases in F* and E*. 3.

Suppose that V = E ®F, where dim F = 1. Let co e /\°(V), and let v e F

such that v 0 0. Show that w= a+ V A fi for unique x e A (E) and 0eAP-1(E).

Boundary value problems for elliptic systems

112

4.

Let E, E' and F, F' be arbitrary finite dimensional vector spaces over R.

Show that the map L(E, E') x L(F, F') - L(E ® F, E' (D F') defined by (f, g) r-+ f ® g is smooth. By f ® g we mean the (f ® g)(v (9 w) = f(v) ® g(w). Hint: The map is bilinear.

map such that

If A is an n x n matrix and IIA II < 1, show that I - A is invertible and has inverse given by the Neumann series 5.

(I-A)-1=I+A+A22+ Use this result to give another proof of the fact that GL(R") is open in L(R").

(a) Let A = A(t) be an n x n matrix where each entry a,, = a,,{t) is a differentiable function of a single variable t. Show that det A(t) is a 6.

differentiable function of t and d dt

det A(t) _ Y A;(t) ;a,

where A;(t) denotes the determinant of the n x n matrix obtained from A(t) by replacing each entry a,,{t) of row i by its derivative (d/dt)(a1,1t)).

(b) Suppose we regard L(R") as R"' with points denoted x = [x,;], then det: GL(R") - R is a map defined on an open set in R" Find a(det)/ax;;. Show that det is a C function. 7.

Let E and F be finite dimensional real vector spaces. Also let gyp: E - R"

and iii: F -+ R' be linear isomorphisms, obtained from a basis in E and in

F, respectively. Show that P. U c E -+ F is of class C' if and only if 0-focp-1:cp(U)cRR -+ R'" is of class C'. 8. Let f,: U e E -+ F,, l < i ,
x Dff(x) Df(x) = Df,(x) x (Leibniz rule) Let U c E be an open set. Let f: U -- F, and g: U -+ F2

be of class C2. If B: F, x F2

G is a bilinear map show that

D[B(f, g)](x) v = B(D f (x) v, g(x)) + B(f (x), Dg(x) v)

where B(f, g): U -+ G is defined by x -+ B(f (x), g(x)). (The usual Leibniz rule is obtained when F, = F2 = G = R and B is multiplication.) Hint: B(f, g) = B c (f x g) where (f x g)(x) = (f (x), g(x)) as in Exercise 4. Now use the chain rule. 10. Let <, > be an inner product (or any symmetric bilinear form) on E. Define the polynomial map f: E - R by f(x) = <x, x>. Show that ID2f(x) _

<, ) 11.

Let f be a map E -+ F. Let e,, ... , e" be a basis and let cp: E -+ R" be

the linear isomorphism defined by cp(y xre;) _ (x,, ... , x"). Show that is a polynomial in the variables f e P4')(E, F) if and only if f = f x,...... ,, (with coefficients in F).

113

Manifolds and vector bundles

12. A map f : E -* F is homogeneous of degree k > 0 if f(tx) = tk f (x) for all t e R and x e E. Show that if f is Ck then it is necessarily a polynomial F) (and hence f is C map, i.e. f e Hint: By Taylor's formula we get

f(h) = k! Dkf(O).h(k) + o(Ihlk)

since Dif(0) = 0 for j < k. Replace h by t - h, divide both sides by tk and let t-1. 0. 13.

Define the function f: 682 - I8 by 2)

f(x,Y)=

xY(x2 - y

if (x, Y) # (0, 0)

x2+y2

if (x, y) = 0, 0)

0

Show that f is of class C'. Conclude from Exercise 12 that f is not of class C2.

A linear differential operator .d of degree k on an open set U in R" has the form ,d (x, a) = I=Y k az(x)

aa alt ... ax." ax"

where A, are Cm real-valued (or matrix) functions on U. For short we write sd = Y,1,1 ,k a,(x) sa, and the function

sd(x. S") = Y a,(x)S2,

a polynomial in

e R",

I2IEk

is called the complete symbol of .d. Note that ml(x, ) = e where denotes the usual inner product of vectors in R". 14.

sd(x, a) e" {

Let sd(x, a) be a linear differential operator on U c I8". Prove

Hormander's generalized form of the Leibniz rule: sd(x, a)(uv) = Y (sd'°`'(x, a)u) a=v(x),

u,

v e C°'(U)

where sd(")(x, S) = 04W(x, ). Hint: If we fix v then each side of this formula can be regarded as a differential

operator in u; show that both have the same complete symbol (use Taylor's formula). 5.2 Manifolds

A locally Euclidean space M of dimension n is a topological space M for which each point has a neighbourhood homeomorphic to an open subset of 61.". A coordinate map is a pair (U, x) where U e M is open and K is a homeomorphism of U onto an open set in R". (Often it is convenient to assume that U is connected.)

114

Boundary value problems for elliptic systems

When dealing with more than one coordinate map we use the notation K: U" -+ V" to indicate the domain U. and range V" c 11" of the coordinate map.

Definition 5.11 Let r be an integer > 1 or oo. A C' structure on a locally Euclidean space M is a family 5 of coordinate maps x: U - V" such that (i) The domains U. cover M: U"E U,, = M. (ii) If K, K' C- a and U. n U,, # 0 then the overlap map U".)

is C'

(iii) The family a is maximal with respect to (ii): if KO is a coordinate map such that Ko o K-' and K - KO ' are C' for all K E t5 then KO E

.

Note: Since K is a homeomorphism the set K(U,, n U",) is open in R". Also, if the overlap map K- K-' is C' then so is its inverse K o K'-' by Theorem 5.8. If ddt = {(U1, K,)}

(i ranging in some index set) is any collection of

coordinate maps satisfying (i) and (ii) then there is a unique C' structure j5 containing 91; namely, let a- = (K-, K is a coordinate map and K1 K - ' is C' for all i}. Then j5 21, and it is easily checked that it satisfies (i) and (ii). j5 is maximal with respect to (ii) by construction. A C' structure can thus be defined by an arbitrary family 21 satisfying (i) and (ii) only. Such a family is called a C' atlas and two atlases are called equivalent if they define the same C' structure. Clearly, two atlases 211 and 212 are equivalent if and only if 91t v 9I2 is an atlas. Definition 5.12 A C' manifold is a pair (M, a) where M is a locally Euclidean,

Hausdorff, second countable space (i.e. there is a countable basis for the topology), and a is a C' structure on M. Usually we refer to the underlying space M as the manifold. The maps in car are called admissible coordinate maps, or charts. If K is a chart with domain

U n x, we say that (U, K) is a chart at x. It is known that any Cr atlas (r 1) on a set contains a C°° atlas; see [Hi]. Thus, it is no loss of generality to assume that r = co, which we do from now on. Examples (1) An n-sphere is the set S" = {x a R"+'; IxI = 1 } with the relative topology as a subset of 18"+'. Let N = (0,. . . , 0, 1) and S = (0, ... , 0, -1) denote the

north and south poles. Then a C' structure on S" is defined by the atlas {K1, K2} where K1: S"\N -+ 18" is defined by

(x1,....X"+1)-+(x1/(1 - x"+1)....sX"/(l -Xn+1)) and K2: S"\S -+ 1%" is defined by (X1,....X"+1)+"+(x1/(1

+X"+1))

Manifolds and vector bundles

115

The overlap map is ice o K1 ': R"\O - R"\O and is given by K2 o Ki '(y) _ y/Iy12, y e R"\O which together with its inverse is C°° so condition (ii) is satisfied.

Remark The charts K1 and K2 are essentially "stereographic projection" from the north and south poles, respectively. For example, the stereographic

x, Fig. 1. Stereographic projection from the south pole.

projection of the point x = (x,, . . . , x"+ ) from the south pole to the plane x"+, = I is the point with coordinates (2x,/(1 + x"+ 1), ... , 2x"/(l + x"+ 1), 1).

(2) R" is a manifold with an atlas formed by a single chart (R", identity). This is called the standard C°° structure on R". Further, if E is a finite dimensional vector space then, by virtue of a linear isomorphism x: E - R" defined by any basis of E, there is an atlas {K} for a C' structure on E. Any two such linear isomorphisms K, K' define the same C°° structure because the overlap map K' a K-': R" - R" is a constant matrix, hence is C.

(3) It is possible for a locally Euclidean space to possess non-equivalent atlases. For instance consider M = R and the following coordinate maps on R: K(t) = t and w(t) = t3. Then 91, = {K} and 212 = {w} are atlases on R but w-' , x is not differentiable at the origin. In other words 211 and 212 define distinct C' structures on R (but they are diffeomorphic, see Exercise 3 at the end of this section.

(4) An open subset U of a manifold M with C'O structure a is itself a manifold with C' structure RU = {K e a; domain of K C U). Note that if K E a and the domain of K intersects U, then its restriction to U also belongs to a by virtue of the maximality condition (iii).

Next we define the concept of a smooth map between manifolds and the local representation of a map.

Boundary value problems for elliptic systems

116

Definition 5.13 Let M and N be manifolds. We say that f: M -+ N is of class C'° (or smooth) if

(i) f : M -+ N is continuous

(ii) for each chart (U, (p) of M and chart (V, 4) of N such that f(U) e V, the local representation, 41, = l, o f - cp - ', is of class C".

The set of all smooth maps f: M -+ N is denoted C'(M, N).

Proposition 5.14 Let 91M be an atlas on M and 21N an atlas on N. Let f: M -+ N be continuous. Then f is of class C°° if and only if for each m e M, there exist charts (U, (p) E `BM and (V, i/i) E 91N such that

m e U, f(U) a V, and the local representation f,, is C'. Proof The necessity is clear. Conversely, if this condition holds, let (U, (p)

and (V, i/i) be any charts of M and N, respectively, such that f(U) c V. We must show that the local representation f 0 is C°°; it suffices to show this in a neighbourhood of each m e U. Now, by hypothesis, there exist charts (U', q,') e 21M and (V', 016 2IN such that m c- U', f(U') c V', and the local representation f.... is C'°. By restricting the domains of gyp,

and 0' we may assume that U = U' and V = V'. Then f,, = (0 o Ji'-') ((P ° V -') -' is C'° since it is a composition of C`° functions defined on open sets in R" (Proposition 5.1). If we recall that the composition of C'* maps on open sets in Euclidean space is C° we obtain the following proposition.

Proposition 5.15 If f e C°°(M, N) and g E C°°(N, P) then go f EC'(M, P).

A map f: M - N where M and N are manifolds is called a diffeomorphism if f is C°°, bijective, and f -': N -, M is C'°. If a diffeomorphism exists between two manifolds, they are called diffeomorphic.

The product of two manifolds is defined as follows. If M, and M2 are manifolds with Cm structures a, and 52 then M, x M2 is a manifold with atlas {K = K1 X K2; KJ E RJ,.l = 1, 2}

where if KJ: U,;j -

c R"' then K, X K2: U,,, X U12 -+ V,,, X VX2 c R"''", is

the product map. The C'° structure for M, x M2 is the maximal family containing this atlas.

An example of a product manifold is the n-torus T" = S' x

x S1 (n

times).

We conclude this section with a study of the axioms for an atlas. Often it happens that an atlas is constructed on a set S which is then used to define a topology on S.

By an atlas on a set S we mean a collection of pairs 91 = {(U;, K;)} (i ranging in some index set) satisfying the following conditions:

Manifolds and vector bundles

117

AT I The Ui's cover S

AT 2 Each Ki is a bijection of U; onto on open set K1(Ui) in R", and for any i, j, Ki(U, n Uj) is open in 18".

AT 3 If Ui n Uj # 0, the overlap map KI

Kj ': K,.(Uj n U;) - K;(Uj n U;)

is C°D

Conditions AT I and AT 3 are the same as (i), (ii) previously. It is necessary

to add condition AT 2 since S is just a set so that the K, are not (a priori) homeomorphisms. Proposition 5.16 Let 21 = {(Ui, Ki)} be an atlas on a set S. Then the collection of subsets of S,

= {W; 3Kj E 21 such that W e Uj and Kj(W) is open in R"},

forms a basis for a topology on S. This topology makes S into a locally Euclidean space and the family 91 is an atlas for a C" structure on this space.

Proof The key point in the proof is the following: if W e `., then Ki(U, n W) is open in 11" for every Ki E'2I

(15)

By definition there exists K E 21 such that W c U,, and K(W) is open in R". Then, for another K, E 21, we have

Ki(Uin W)=Ki°K-'(K(Uin W)) which is open in 68" by virtue of the Inverse Function Theorem since Ki c K is a diffeomorphism and K(U, n W) = K(Ui n UK) n K(W) is open in R".

Let W, W' e G. Then for some Ki, Kj a 21 we have W c Ui, W' (-- Uj and Ki(W) and K/ W') are open in R". If we can show that W n W' e S, then G is a basis for a topology on S. But this follows since W n W' c Uj and

K,(Wn W')=K Uin Uj)nKj(W')nK,{Ujn W) which is an intersection of three open sets in R" (see (15) for the third set) and hence is open itself. We claim that with this topology each K e 21 becomes a homeomorphism. By the definition of C it is immediately obvious that K is continuous. It remains to show that K(U) is open in R" for every open set U in the domain of K. But since U is a union of sets in Z it follows that

K(U)=U{K(W); WE C., We U} is open since it is a union of sets K(W) which are open by (15). Hence K is a homeomorphism. Thus every point in S has a neighbourhood homeomorphic to an open set in R". Remark The set S with the topology in Lemma 5.16 is not necessarily second

countable, or even Hausdorfl (see Exercise 8 below). In other words, we cannot conclude that S is a manifold in the sense of Definition 5.12.

Boundary value problems for elliptic systems

118

Exercises

Show that the upper hemisphere S+ = {x E 18'; IxI = 1, x3 > 0} of the 2-sphere is diffeomorphic to the open disc D = {z e R2; JzJ < 1), where S+ and D are given the C'° structures as open sets in S2 and 182, respectively. 1.

2.

Let U,-' _ {x E S"; ± x; > 0} and define x; : U; -+ {x' a 68"; Jx' J < 1 } by

XI-- X =(XI,...,xi-1,X,+,...... n+1) Show that 21 = {(U; , xi is an atlas of 2n + 2 charts for S", and that the C' structure defined by 21 is the same as that defined by the atlas in Example (1). 3. Consider the two C" structures on F in Example (3). Show that the map t -+ t`f3 is a diffeomorphism from (68, a,) to (R, j2).

Let B,(xo) denote the open ball in U8" of radius r and center x0. Show -jxJ2)-12 is a diffeomorphism from B1(0) to Ui". Conclude that any manifold has an atlas {(U,, h,)} for which all h,(U,) = ff8". 4.

that the map 5.

(a) For fl: M --> Ni, i = 1, 2, define f: M --' N, x N2 by f(m) = (f1(m), f2(m)).

Show that f e C°`(M, N, x N2) if and only if both f e C'°(M, Ni), i = 1, 2. (b) For g,: M, - Ni, i = 1, 2, define g, x g2: M, x M, --+ N, x N2 in the obvious way. Show that g, x g2 is of class C' if and only if g, are of class C°`,

i=1,2.

Quotient topology

An equivalence relation on a set S is a binary relation x, y and z e S:

such that for all

(i) X - X

(ii) x-yandy-x

(iii) x -- yandy - zimpliesx - z The equivalence class containing x is [x] _ {y e S; x - y}. The set of equivalence classes is denoted S/ - and the map n: S -, S/ defined by x )-+ [x] is called the canonical projection. If S is a topological space, the collection of sets

{U c S/-'; n-(U) is open in S} satisfies the axioms for a topology, called the quotient topology on S/ -. Also see the text at the end of §5.4 for a discussion of quotient manifolds. 6.

Let - be an equivalence relation on a topological space S and

n: S - S/-i the canonical projection. Let T be another topological space. Prove that a map q : S/- T is continuous if and only if W e n: S T is continuous.

Manifolds and vector bundles 7.

119

The set r _ {(x, x'); x - x'} c S x S is called the graph of the equival-

ence relation -. Note that f = or x ir) '(A) where A is the diagonal of

(S/-) x (S/-).

(a) If S/ - is a Hausdorff space, show that r is closed in S x S. (b) Conversely, if r is closed in S x S and it is an open map (i.e. the image of any open set in S/- is open in S), show that Si- is Hausdorff.

(line with two origins) Let S = (R x 0) u (R x 1) and define the equivalence relation - on S which identifies (t, 0) with (t, 1) for t 0 0. Let 8.

h;:Rxi-+6Fbedefined asx;(t,i)=t for i = 0, 1. (a) Show that 'U = {(R x i, h,)},v0., is an atlas on Si-. (b) Show that the induced topology on Si-. (by Proposition 5.16) is locally Euclidean but non-Hausdorff. (c) Show that the topology in (b) is the quotient topology.

The real projective space RP" is defined as S" (the unit sphere in R"+') with antipodal points x and - x identified for all x e S". The canonical map n: S" -+ RP" is given by x t- [x] = {x, -x}. (a) Show that RP" with the quotient topology is a compact, connected and Hausdorff space. (b) Show that the restriction of the canonical map n to any open hemisphere in S" is a homeomorphism onto an open set of RP". Conclude that RP" is locally Euclidean. (c) Show that there is a unique C' structure on RP" such that a restricted to any open hemisphere is a diffeomorphism, and with this structure RP" is a manifold. 9.

5.3 The tangent bundle Let M be a manifold and m e M. In this section we define the tangent space, 7,,M, at the point m and show that the union, TM, of all these tangent spaces

itself forms a manifold, called the tangent bundle. Then we show that a smooth map between manifolds induces in a natural way a map between their tangent bundles. Let 1 denote an open internal in R with 0 E 1. A curve at m is a C r map c: I -+ M such that c(O) = in. We say that two curves c, and c2 at in are tangent at m if they have identical tangent vectors in every chart, i.e. (K ° c 1)'(0) = (k ° c2)'(0)

(16)

for every chart (U, K) with m e U (we denote the derivative of a function g: I -+ R" at t = 0 by g'(0) e R"). The intervals 1 j on which c j are defined are of course taken sufficiently small so that c fflj) lies in the domain of a, j = 1, 2. Tangency of curves at m e M is a concept that is independent of the chart

used, i.e. it suffices that (16) holds for one chart K at m. Indeed, let is be another chart at m. Then if we multiply (16) by the matrix D(i: ox'')(>;(m)), it follows by the chain rule that (i: 0 c,)'(0) = (ic o c2)'(0) also.

120

Boundary value problems for elliptic systems

Tangency at m e M is an equivalence relation among curves at m. An equivalence class of such curves is denoted [c]m, where c is a representative of the class. Definition 5.17 The tangent space to M at the point m is the set of equivalence classes of curves at n TmM = {[c]m; c is a curve at m}

The disjoint union of the tangent spaces is the tangent bundle

T.M = U TmM meM

To emphasize that this is a disjoint union we often write elements of TmM

in the form (m, t) where t = [c]m. The map n: TM -+ M defined by n([c]m) = in is the tangent bundle projection. If U is an open subset of M, it inherits a CI structure and we identify TmU = TmM for m e U. Then TU = n 'U = (J,, TmM with projection nu: TU -+ U. The set of smooth real-valued functions f: M - R is denoted C'(M, R), or just Cm(M) for simplicity. If f e C'°(M) we define the differential of f at m e M to be the map df(m): TmM -+ R

defined by [c]m i--+ (f o c)'(0)

which is well-defined, independent of the representative c, by virtue of the chain rule. Then we define the map df: TM -+ I8 such that its restriction to TmM is d f (m), that is, d f I T.,,At = d f (m). We will often write d f Im instead of d f (rn), using whichever notation is convenient.

Remark

(i) If U is open in M then d(f l u) = df I ru (ii) Leibniz' rule holds for f, g e C"(M) d(f-g) = f-dg +

that is, d(f

f (m) dg(m) + g(m) d f (m) for all in e M.

If P U -+ l is vector-valued, then df is defined in the same way as above.

Note that if we write f = (f,..... fk) then df = (df,,...,dfk): TU

R".

Theorem 5.18 TmM has a unique vector space structure such that, for any function f e C x (U, R), where U is an open neighbourhood of m, the map

df: T,M

1k is linear. (Note: When no confusion is possible we write d f instead of d f(rn).)

Proof If (U, K) is a chart at m e U, then K: U -+ fl" and we have a map dK: TmM -+ iB". By the definition of dK and of tangency of curves it is clear that dK is injective. We claim that dK is bijective. For v a R" define the curve c(t) = K''(K(m) + tw), t e (-S, S); b is sufficiently small. Then dK[c] = (K - c)'(0) = w which shows that dK is surjective. Since dK is bijective

we can use it to impose a vector space structure on TmM; in other words

Manifolds and vector bundles

121

such that

dK([cl]m ®[c2]m) = dK[C1]m + dx[C2]ml

(17)

a 68 J = This definition of the vector space operations ® and O in T,M is inde-

dK(i.O [ c ]

pendent of the choice of chart at in, for if (V, c,) is another chart at n: then multiplication of the equations (17) by the map D(4po K-')(K(m)): Q8" - 88, which is linear, shows via the chain rule that drp([c1]m ® [c2]m) = do[c1]m + dq [c2]m

(17')

d4 (%O [C]m) _ %-dw[c]m, A E R

Thus the equations (17) hold if and only if (17') holds and the vector space operations on T,M are well-defined. Now let f: U -+ I8' be a smooth map. We wish to show that df: T,M R't is linear. Let [ct]m e TmM, j = 1, 2 and let Cc]., = [c,]. ® [c2]m. Then by definition of df and applying the chain rule it follows that dJ ([c1]m ® [c2],,) _ (f -c)'(0) = D(f c K -')(K(m)) (K ° c)'(0)

By virtue of the definition (17) of [c]m = Cc,]. ®[c2Jm, we have (K o c)'(0) = (K o cl)'(0) + (K o c2)'(0) for any chart K at in, hence the linearity of D(f K-')(K(m)) implies df([C1]m (D[C2]m)

= D(f ° K -')(K(m)) '(K ° C1)'(0) + D(f °

K-')(K(m)).(K

° c2)'(0)

= (f °cl)'(0) + (f °c2)'(0) = df[C1]", + df [C2]m

Similarly, one shows d f(;. O [c]m) = ). (f o c)'(0) = ). d f [c]m for all i. 6 R. Hence df: TTM -. R1 is a linear map.

The differentials, dK, make it possible to define a topology and a C' structure on the tangent bundle. If (U, K) is a chart, we have the map TK: TU - K(U) x R" given by [C], h- (K(m), (K ° c)'(0))

that is, TK = (K o n, dK) where n: TU - U is the tangent bundle projection. As shown above, dK(m): T,M W' is a bijection for each in, whence TK is a bijection from TU to K(U) x 18". Now we can use Proposition 5.16 to prove the following theorem. Theorem 5.19 The family 91 = {(TU, TK); (U, K) is a chart on M }

is an atlas for a C' structure on TM. making TM into a manifold. With this structure, then for any C' map f: M -+ R*, the map df: TM - l is also C. The tangent bundle projection n: TM -+ M is a Cm map. Also M e TM

Boundary value problems for elliptic systems

122

if we identify m e M with the zero vector 0. E T,"M, and this makes M into a C°° submanifold of TM.

Proof The conditions AT I and AT 2 obviously hold (Proposition 5.16). To verify AT 3, let (U, K) and (V, (p) be two charts for M. If x e ic(U n V), w E R",

we have (TK)-'(x, w) = [c]," whence c is a curve at m = K-'(x) such that (K a c)'(0) = w. Then Tcp a (Tic)-'(x, W) = Tq)([c].) = ((P(m), ((P a c)'(0))

=

(4P(K-1(x)),

D((o a K-1)(Y) - w)

which is a C°° map from K(U n V) X R" to q (U n V) x R". This shows that 91 is an atlas for a C°° structure on TM. It is not hard to show - with the topology on TM as defined in Proposition 5.16 - that TM is Hausdorff and has a countable basis since this is true of M and R n. The proofs of the last three statements in the theorem are left to the reader. Remarks (i) If U is an open subset of a manifold M then it is itself a manifold in the obvious way, and T.U and T,,M can be identified for each m e U.

(ii) There is an identification TR" - R'` x Rk given by the collection of maps {x} x Rk defined by [c]., )-4 (x, c'(0)). from

Definition 5.20 Let f: M - N be a C°` map. We define Tf: TM -p TN by Tf([CJm) = If aCJf( )

Tf is called the tangent of f. The notation T. f is also used for the restriction of Tf to the tangent space T",M; we regard T,,f as a map T",M - Tf(M)N.

TA C). = If ° C)l(m)

Fig. 2. The tangent map Tf.

Manifolds and vector bundles

123

Note: df is defined only if f is real-valued (or vector-valued) on M. whereas the tangent map, Tf, is defined for any C map f between two manifolds and Tf is a map between the tangent bundles. By virtue of Remark (ii) above we see that if f: M -+ Rk is a C°o map then Tf([c]m) = (f(m), df [c]m)

Cc]m a T,M

(')

for any f: M - N. Indeed, if (U, K) and Locally, Tf has the form (V, gyp) are charts on M and N, respectively, such that f(U) c V then Tip o Tf o (TK) -' is given by

K(U)x R"

rp(V) x R°

(n =dimM,d= dim N)

(x, W)-(((P-f° K-')(x), D(q o f o K-')(x)' w), for if c is a curve at m = K-'(x) such that (K o c)'(0) = w then

(18)

Top- TI c(TK)-'(x, w) = Trp° TfCC]., = Tco[f oc]t(m) = (w(f(m)), ((P of oc)'(0))

= (((pof oK-')(x), D(cpof oK-')(x)-w) Theorem 5.21

(i) 1ff:M-+NisaC"'

(ii) Suppose f: M - N and g: N - P are C" maps of manifolds. Then gof: and Tigof) = TgoTf (iii) 1ff: M -+ M is the identity map, then TI: TM -+ TM is also the identity.

(iv) 1f f: M - N is a diffeomorphism, then Tf: TM - TN is a bijection and

WY' = 7 f ').

Proof The statement (i) follows from (18) since lp o f o K-' is smooth by definition (see Definition 5.13). By using local representatives it is easily seen

that g o f is C°° if f and g are C°° (see Proposition 5.15). Moreover, and

T(gof)[c]., = [gof oc]gafirm

(Tgo Tf)[c]m = Tg([f °C]t(m,) _ [gof whence T(g o f) = Tg o Tf.

(iii) is obvious, and (iv) follows from (ii) and (iii).

Remark The following chain rule holds for smooth functions g: N -+ R and

f:M-+N,

d(gof)=dgoTf

(19)

Exercise Let M, and M2 be manifolds and pl: M, x M2 - M, and P2: M, X M2 _+ M2 the canonical projections. Show that the map (7p Tp2) is a diffeomorphism

124

Boundary value problems for elliptic systems

of the tangent bundle T(M, x M2) with the product manifold TM, x TM2.

With this identification, show that if g,: N -+ M, and g2: N - M. are smooth maps then T(91 x 92) = T91 X T92-

5.4 Submanifolds

Let us write R" = R"-" x R" and denote points of R" by x = (x', x") where

x' c- R", x" a R". The notation R"-" x 0 indicates the subspace of R" consisting of points (x', 0, . . . , 0) (k zeros). An (n - k)-dimensional submanifold of an n-dimensional manifold M is a

subset Y c M endowed with the topology induced from M and with the property that for each y e Y there is a chart (U, K) of M with y e U such that SM

K(U n Y) = K(U) n (R"-" x 0)

(the submanifold property for K). The number k is called the codimension of Yin M.

An open subset of M is a submanifold of M of codimension 0. A submanifold of codimension I is called a hypersurface of M.

Proposition 5.22 Let Y be a submanifold of a manifold M, and give Y the relative topology induced from M. Let {(UU, Ki)}ie, be a family of charts satisfying the SM property and such that the sets U, n Y cover Y. If we identify

R"-" with the subspace R"-" x 0, then the family of pairs {(U1 n Y, Kilu,ny)}iet

is an atlas for a C" structure on Y, making Y into an (n - k)-dimensional manifold. The inclusion map i : Y - M is C'. Proof Any coordinate map K: U - K(U) (i.e. a homeomorphism to an open U n Y -, K(U n Y) set in R") on M restricts to a homeomorphism is a coordinate map on and, further, if K satisfies property SM then Y (i.e. a homeomorphism to an open set in R"). We may let K = Ki for any i e 1, and the open sets U, n Y cover Y, thus it follows that Y is locally Euclidean. To complete the proof we must show that the overlap of two maps, (Ui n Y, Kilu,ny) and

(Uj n Y, Kjl ujnY),

is C''. But this is clear since K j o Ki ' is a C'° function defined on an open set in R" and

Kjlu,r,r°(K"lu,nr)-' = Kj0Ki 'IKi(Ui n Y)

is a restriction of this C' function to an open set in R"-". Hence (ii) of Definition 5.11 holds and 91 is an atlas for Y. Also Y is Hausdorff and has a countable basis since this is true for M.

Manifolds and vector bundles

125

The inclusion map i is C" for if (U, K) is a chart on M satisfying the SM property then the local representation of i with respect to charts K and fl" and is hence smooth. KIU,,r is a restriction of the inclusion 12"-k Example Consider the set Y c RI as in the figure below:

p

Fig. 3. A manifold which is not a submanifold of R2.

The arrow means that the line approaches itself arbitrarily close at p, without touching. It is clear that there is a bijection R - Y. By means of this bijection

one can define a topology and C'° structure on Y; however, Y is not a submanifold of RI because the topology on Y is not the relative topology induced from 182. (To show this, examine neighbourhoods of the point p.) Submanifolds have the following universal property. Proposition 5.23 Let Y be a submanifold of M. Given another manifold N, and

a C' map f: N - M such that f(N) e Y, let fl: N - Y denote the induced map. Then f, e C'°(N, Y).

Proof Note that in the simplest case when Y = 68"-` x 0 and M = R", the proposition is obvious (see §5.2, Exercise 5). The same is true when Y = W n (Q8"-" x 0) and M = W is some open set in R". The SM property enables us to reduce the general case to this case. Suppose that f is C'. Then f is continuous, and since Y has the relative

topology in M it follows that f, is continuous. To show that f, is C" it suffices to show it is smooth in a neighbourhood of each x e N. By property SM there is a chart (U, K) at y = f(x) such that K(U n Y) = K(U) n (R"'" x 0). Then there is a neighbourhood V a x such that f,(V) c U n Y. The function Kof,Iv maps into K(U) n (Q8"-I x 0), so it is smooth since K o f I,, is smooth.

Corollary The universal property in Proposition 5.23 uniquely characterizes the C' structure on Y.

Proof If ji, and j52 are any two such C' structures on Y then this property implies that the (set-theoretic) identity map Y - Y is C°° from (Y, &) to (Y. a2). Hence O2 a a,, for by composing a chart in a2 with the identity it becomes a chart in &. Similarly, a, a j52, so that Fr, = j52. The structures are the same, not just diffeomorphic.

126

Boundary value problems for elliptic systems

For examples of submanifolds, see the Submersion Theorem below.

Let us now consider the tangent space to a submanifold Y c M. The inclusion map 1: Y - M is smooth, and we claim that the tangent map T.i : T.Y -+ T,.M,

y c Y,

is injective; in this way the tangent space, T, Y, can be identified with a subspace of T,.M.

To prove that the tangent map is injective it suffices to work in local coordinates. Let (U, K) be a chart having the SM property. With respect to charts Klur,y and K, the local representation of : is K e o (Klu,,r)-' and is just a restriction of the inclusion f8"-k - R". By Theorem 5.21 we see that the local representation of the tangent map Ti: TY - TM is

TKoT1o(TKIur,r)-' and is a restriction of the inclusion R' -k x R"-k R" x R" (also see (18)). It follows that the tangent map Ti: T,. Y TM is an inclusion for each

yeY. Remark With this identification (of T,. Y as a subspace of T, M), we have

T.Y = (T

K)-'(R"-k

x 0)

for any chart (U, K) at y satisfying the SM property. In the following lemma, E and F denote finite dimensional vector spaces.

Lemma 5.24 Let f: U - F be of class C' where U e E is open. Suppose that, for some uo E U, Df(uo) is surjective. Let E, = ker Df(uo). Then there exists an open subset U' of U containing uo and a C' di,Q'eomorphism 0: W, x W2 --. U'. where W, a E, and W2 e F are open, such that (f °W)(x1, x2) = x2

for all (x1, x2) E W, x W.

Proof Let E, = ker Df(uo) and choose a complement E2 such that E = E, ® E2. Then the partial derivative D2 f(uo): E2 -. F is a linear isomorphism. Thus, we may assume that E2 = F and E = E, x E.

Define the map g: U - E1 x E2 by g(xt, x2) = (x j(x x2)) and note that the derivative of g at uo is the linear map E1 x E2 Dg(uo) =

E, x E2 given by

0

IEt

Dlf(uo) D2f(uo)

which is an isomorphism. By the Inverse Function Theorem, g is a diffeomorphism of some open set U' containing uo onto an open set W1 x W2 containing g(uo). Let 0 = (glu.)''. If (x1, x2) a W, x W2 we have

(x,, x2) = g(0(x,, x2)) and hence 'I'

(x1, x2) = (x,, f(W(x,, x2))) Thus f(W(x1, X2)) = x2

Manifolds and vector bundles

127

Let f e C'(M, N). A point v e N is called a regular value if for each x e f -'(y), the tangent map Tf is surjective. A critical point is a point x e M where Txf is not surjective. If Tx f is surjective for each x in a set S, we say that f is a submersion on S. Theorem 5.25 (Submersion Theorem) Let

f: M -+ N be of class

C°`

and let yo e N be a regular value. Assume that the level set f -'(yo) _ {x; x e M, f(x) = yo} is non-empty. Then f -'(yo) is a submanifold of M with tangent space at x e f -'(yo) given by ker Tx f.

Proof Let xo e f -'(yo). Let (V. (p) be a chart at yo and (U, K) a chart at xo such that f(U) c V. Consider the local representation ffK = 0-f- K_'. We have that T(fQK) = TV c Tf a (TK) -' is the map K(U) x R" - O(V) x Rd given by (x, w) H (fvK(x) D(ffK )(x) - w)

(see (18)). Since Tx"J is surjective, it follows that the derivative is a surjective map. Hence we can apply Lemma 5.24. Let n = dim M and d = dim N. By shrinking U and V, if necessary (and replacing K by 0 -' 0 K), we may assume that K(U) = W, x W2, [p(V) = W2 where W, c R' -d and WZ Rd are open, such that ffK is the projection W, x W2 -+ W. on the second factor.

We also assume that K(xo) = 0. With this choice of charts. we have K(U (I f -'()1o)) _ (f,.)-'(0) = W, x 0, hence

;(U r) f-'(yo)) = ti(U) n (R"-d x 0), which is the submanifold (SM) property. Hence f -'(),o) is a submanifold of M of codimension d = dim N. We also have for u e W, x 0 R"-d x 0 = ker(T"f9K)

where c(x) = u, implies

which in view of TK(f'K) = Tf,x)(p o TXf

that T-x(f -'(3b)) =

(Txn)-'(f8"-d x 0)

= ker T;J.

Example Let f: IB"+' - R be defined by f(x) = x12. By identifying 7 W' OB"' x 68"" 1, we get Tf(x, v) = (Ix12, 2<x, v>)

where < , > is the usual scalar product of vectors in f1"+'. Let x e S", i.e. f(x) = 1. Then TxJ: Tx08"+' - T,08 is the map v i- 2<x, v> and is surjective since x # 0. Hence I is a regular value of f, so the n-sphere S" = f ()) is a submanifold of R"+', and for any x e S" we have Tx(S") = {v;veR"+',<x,v> =0}. We conclude this section with a study of quotient manifolds (see also §5.2, Exercises 6 to 9).

Boundary value problems for elliptic systems

128

Definition 5.26 An equivalence relation - on a manifold is called regular if the quotient space Ml- has a manifold structure such that the canonical projection n: M - M/- is a submersion.

If - is a regular equivalence relation, then M/- is called the quotient manifold of M by -. Proposition 5.27 Let - be a regular equivalence relation on M.

(1) A map f: M/- - N is CD if and only if f' n: M N is C'. (ii) Any C" map g: M -+ N which factors through the equivalence relation

-, that is,

x-y

g(x) = g(1'),

defines a unique C" map g: M/- -, N such that g o n = g. Proof (1) If f is C", then so is f o a by Proposition 5.15. Conversely, let f o it be C'. Since n is a submersion, it can be locally expressed as a projection,

i.e. (as shown above) there exist charts (U, K) at m e M and (V, tp) at n(m) a MI- such that K(U) = W, X W2, qp(V) = W2 where W, a R"-d and W2 c R' are open, and

n,x: W, x W. - W2 is the projection on the second factor. Hence if (V', qi) is a chart at f on(m) in N satisfying f o n(U) c V', then

=O°(f°n)°K_'1(OIxW2.

Therefore f,,, is C°°, being the restriction to {0} x W2 of the C' function

(f

(ii) It is evident that g is uniquely determined; and it is C' by (i). Corollary 5.28 The manifold structure of Ml- is unique.

Proof Let (M/-), and (M/-')2 be two manifold structures on M/^- having it as a submersion. If we apply Proposition 5.27(ii) to the map n: M -+ (M/-)21 it follows that the set-theoretic identity map (M/ - ), (M/-)2 is C'. But of course the roles of (M/-), and (M/-)2 can be reversed, so the identity map induces a diffeomorphism. There is in fact a bijective correspondence between surjective submersions and quotient manifolds; see Exercise 6. Exercises

(a) Show that if Y, is a submanifold of M;, i = 1, 2, then Y, x Y2 is a submanifold of M, x M2. (b) Show that if f: M N is a diffeomorphism and Y c M is a submanifold then f (Y) is a submanifold of N. 1.

Manifolds and vector bundles

129

Let h: M -+ N be C°`. Show that if Y is a submanifold of M then the restriction hl r: Y -+ N is C'. Hint: The inclusion map i: Y - M is C'. 2.

Let Y be a submanifold of M. Show that f e CO'(Y, k) if and only if for every y e Y there is an open set U in M and a function f e C'(U, N) such that y e U and 3.

Ilvr-r=fIvr-r 4.

Verify that the C' structure on S" defined in the example above (by

means of Proposition 5.22) is the same as the one defined by stereographic projection.

Let f e C'(M, N). The graph off is defined to be graph(f) = {(x, f(x)); XEM}. (a) Show that graph(f) is a submanifold of M x N of codimension k = dim N. 5.

(b) Show that the inclusion map, graph(f) -, M x N. induces a diffeomorphism ftmuu(graph(f )) = graph(T," f ). Hint for (a). First consider the case N = 68`. Define the map gyp: M x N -

M x N by (p(x, y) = (x, y - f(x)) and use the Inverse Function Theorem to show that (p is a diffeomorphism of a neighbourhood of (x0, f(xo)) onto a neighbourhood of (x0, 0). 6.

Let f: M -' N be a submersion and let - be the equivalence relation

defined by f, i.e.

x - y. f(x)=I(Y) Show that - is regular, M/- is diffeomorphic to f(M), and f(M) is open in N.

Let M and N be manifolds, and R and Q regular equivalence relations on M and N, respectively. If f: M - N is a C" map compatible with R and Q, that is, if 7.

xRy - f(x)Qf(y), then f induces a unique C°° map J: M/R - N/Q such that nN, f = J O ltM.

A C" map f: M -+ N is called an immersion at m if the linear map T", f. T,,M - T",N is injective. If f is an immersion at each m, we just say that f is an immersion. A C' map f: M - N is called an embedding if it is an immersion that is a homeomorphism of M onto the image f(M) with the relative topology induced from N.

Let f: M -, N be a C°° map. Show that the following are equivalent: (a) f is an immersion at m. (b) there are charts (U, K) and (V, 0) on M and N, respectively, with 8.

f(U) c V, m E U, K(m) = 0, K: U -+ U' and 0: V - U' x V' such that

0 o f e K-1: U' - U' X V' is the inclusion x i-4 (x, 0). (c) there is a neighbourhood U of m such that f(U) is a submanifold in N and f restricted to U is a diffeomorphism of U onto f(U). 9.

Find an example of an injective immersion which is not an embedding.

130

Boundary value problems for elliptic systems

10. Let N be a manifold. Show that a subset S c N is a submanifold if and only if S is the image of an embedding.

5.5 Vector fields

A vector field on a manifold M is a C'O map X : M - TM such that X. a T. M

for all m e M (i.e. a smooth section of the tangent bundle TM). The set of vector fields on M is denoted by 1(M). C'(M) is a commutative ring under pointwise addition and multiplication of functions, and X(M) can be given a C"'(M)-module structure as follows: if X, Ye f(M) and f e C'(M) then X + Y E AC(M) and f X e Y(M) are defined by and (fX)(m) =

(X + Y)(m) = X(m) + Y(m)

To facilitate calculations in local coordinates, we introduce some notation.

Let i.': R" -. R denote the projection on the ith component. If (U, K) is a chart for M, we write K =

K: U - R are the component functions of K. Note that xi is that is, x; Cm since A' and K are C°°. Hence dx;: TU - R is C1, and, by Theorem 5.18,

dx,)m a (T,,M)*, a linear function on T,"M for each m e U. We have dK = (dx1, ... , dx") so the definition of TK takes the form TK(m, v) = (K(m), dx1(v),... , dx"(v))

for v e T.M. Let e1, ... , e" denote the standard basis of R" and let c/axjI," denote the unique vector in T",M such that dKIm(c/exjlm) = ej, that is, dxd.(a/ax j1",) = b j

i, j = i, ... , n c/cx"l," form a basis of T,,M and

For each me U, the vectors

dx1I" .... dx"J. is the corresponding dual basis in (T,,M)*. Note that c/11x...... c/ex" are vector fields on U since TK o a/ax j o K -1(x) = (x, e j ).

X E K(U ),

(20)

is C' on K(U), whence c/cxj: U - TU is smooth. Proposition 5.29 Let (U, K) = (U, x 1..... x") be a chart on M. Then X(U) is generated by c/11x1, ... , e/11x" as C°`(U)-module, i.e., a map X: U - TU is a vector field on U if and only if it has the form

X=X1

0 ex1

+...+Xn ex"

where X' e C'(U), i = 1, ... , n.

Proof Let X be a vector field on U. Since X(m) a T, M, the definition of

Manifolds and vector bundles

131

a/ax,l. implies that X(m) _ P ( M ) ax

I

+...+ X '(M)

Im

(21)

8x "

where X'(m) = dx,(X(m)). The X' are smooth since X' = dx; a X where v: U - TU and dxr: TU - R are smooth. Conversely, any map X: U -' TU of this form is obviously smooth and X(m) E T",M for all m e U; thus it is a vector field on U. Let X e X(M), a vector field on M. The directional derivative of f E C°°(M) along X is defined to be Xf := d f o X, that is,

(Xf)(m) = df(m)(X(m)).

Since d f and X are smooth then so is Xf. See Exercise 2 at the end of this section for properties of the directional derivative. Now let (U, K) = (U, x ... , x") be a chart on M. For the vector fields

0/ax; E 1(U), it follows by the chain rule that the directional (partial) derivative (a/ax;) f= d f o a/ax; is given by a,{ f o K- I)(K(m)),

ax f (m) =

ME U,

where f e C'(U) and a; is the jth partial differentiation operator for functions defined on open sets in R". Then we have

f

d f (m) = o

(m) dx I (m) +

- + az (m) dx"(m),

meU

(22)

"

1

Now we turn to the definition of pull-back and push-forward for functions and vector fields. If cp: M - N is a diffeomorphism and X E f(M), a vector field on M, the push forward of X by rp is defined by (p*X = Tcp e X ° cp- I E 1(N)

(23)

It is the vector field that makes the diagram

T(M) T*

T(N)

xtl

M commutative. Note: If x e N then ((p*X )fix a TTN, so the push-forward of X is indeed a vector field on N. We also define a pull-back map qp* for vector fields by replacing (P by gypthat is, (p*X = Tcp- I o X a cp.

For functions, the pull-back map rp*: C'(N) - CX(M) is defined by (p*(f) _ f o gyp,

.f e C"(N)

Similarly, the push-forward for functions is defined by cp.(f) = f a cp-

Boundary value problems for elliptic systems

132

Remark (i) We have cp* = ((p-')* for both functions and vector fields. (ii) The pull-back of functions, cp*(f) = f ^ cp, is defined for any smooth map

(p, but the pull-back of vector fields makes sense only for diffeomorphisms (but see Exercise 4). The push-forward and pull-back maps are U8-linear on both functions and vector fields. Also, on functions, the pull-back and push-forward maps are algebra homomorphisms, that is, (p*(fg) = (p*(.f).qp*(g)

f, g c- CA(N)

W*(fg) = W*(f)'w*(g)

f, g E C°°(M)

and

The local representation of a function f on M with respect to a chart (U, K) is

f= fok The local representation of a vector field X on M with respect to the charts (U, K) on M and (TU, Tic) on TM is

X=TKoXoK-' (compare with (20)).

Integral curves

Recall that we say c: I - M is a curve at m if I is an open interval, 0 e I, and c(0) = m. Since I is an open set in R we can identify TI = R for each t e I in a canonical manner, and the tangent map Tc: R - T(,,M enables us to assign a tangent vector to the curve at each point c(t) by c'(t) = T,c(1)

Note that if cp: M - N is a smooth map, then

An integral curve of a vector field X e .1(M) is a curve c: I - M such that c'(t) = X(c(t))

for all t e I; and if it also has the initial value c(0) = m, we say that c is an integral curve of X at m. The local representation of c with respect to a chart (U, K) at m is the curve c = K a C in R". Applying the map T(,)K to the equation

c'(t) = X(c(t)) we obtain Z'(t) = X(O(t))

(24)

where X = Tic a X o K-' is the local representation of the vector field X with respect to the charts (U, K) on M and (TU, TK) on TM. Then equation (24)

Manifolds and vector bundles

133

is a system of ordinary differential equations in n unknowns

dc' dt

dc"

dt

= X'(c'(t), ... , c"(t))

= X "(c'(t), ... , c"(t))

where e = (c', ... , c") and k = (X', ... , X"). Thus, for existence of integral curves we may use the following well-known

result on systems of ordinary differential equations. The most difficult part

of the proof is the smooth dependence of the solutions c on the initial conditions c(0); the proof can be found in [AMR]. Theorem 5.30 Let U be an open set in RB" and let X: U - R8" be of class C'. For each x0 a U, there is a curve c: 1 -+ U at x0 such that

do/dt = X(c(t)) for all t e 1. Any two such curves are equal on the intersection of their domains. Furthermore, there is a neighbourhood U0 of xo, a real number a > 0 or a = oo,

and a C' mapping

F: U0xI.-+R8", where 1" is the open interval (- a, a), such that F(x, 0) = x and the curve 1" -+ R" given by t -+ F(x, t) satisfies the differential equation (*).

This theorem is local in nature; it can be applied, essentially without any further proof required, to manifolds.

Theorem 5.31 Let M be a manifold and X e '(M). For each mo e M there exists a neighbourhood Uo of mo, a real number a > 0 or a = oo, and a C°° map

F: such that F(m, 0) = m and the curve 1" - M given by t i-+ F(m, t) is an integral curve of X. If F': Uo x 1,. - M is another map with the same properties then

F and F' are equal on (U0 n Uo) x (I. n IQ ).

Proof Let (U, x) be a chart at x0, and let X : K(U) - R2" be the local representative of X, given by X = TK o X c K -'. By Theorem 5.30 there exists

a neighbourhood Uo c rc(U) and a smooth function F: Uo x I -+ R" such that F(x, 0) = x and the curve t i- F(x, t) satisfies the equation (24).

By shrinking Uo and I we may assume that F(UO X I),e U. Now let U0 = rc-'(U0) and define F: Ua x I -+ M by F(m, t) = h-'(F(x(m), t)). Since c is an integral curve of X if and only if K o c is an integral curve of X, it is clear that F satisfies the required properties.

134

Boundary value problems for elliptic systems

Now let F be another function with the same properties as F. Due to the uniqueness of integral curves proved below in Theorem 5.32, for each x e Uo n Uo we have Fl,,. F'I;x, x 1 where 1 = la n IQ.. Hence F = F' on

(U0c Uo)x1.

Observe that solutions c: I - M of c'(t) = X(c(t)) remain solutions under translations; that is, if t, e I then a(t) = c(to + t) is defined in a neighbourhood of t = 0 and 7'(1) = c'((o + t) = X(c(to + t)) = X(x(t))

This fact allows us to extend the uniqueness part of Theorem 5.31 to get a global uniqueness theorem on manifolds. It is important here that we have required that all manifolds be Hausdorff. Theorem 5.32 If c, and c2 are integral curves of X at in e M then c, = c2 on the intersection of their domains. Proof It is not sufficient just to refer to Theorem 5.30, since we do not assume that the domains of c,: I, -+ M and c2: 12 . M lie in a single chart. However,

observe that the intersection, I = 1, n I2, of the domains of c, and c2 is a connected set. The set J = IS a 1; c,(s) = c2(s)) is certainly closed since M is Hausdorff. On the other hand, J is also open, for if s e J then the curves t'- c,(t + s) and t F-+ c2(t + s) are integral curves of X passing through c,(s) = c2(s) when t = 0, so by virtue of Theorem 5.30 they coincide in some

neighbourhood oft = 0. Hence c, = c2 on a neighbourhood of s. Thus J is both open and closed in I, which implies J = I since I is connected. Corollary Let c,: 1, - M and c2: 12 - M be integral curves of X and suppose that c,(t,) = c2(0) for some t, e I. Then the curve t F- c,(t + t,) is defined on the open interval 1, - t, (which contains 0) and must coincide with c2 on the intersection (11 - t,) n I2 since both are integral curves for X at e2(0). Hence c(t) --

CIO),

fc,(t - t,),

t e 11

t e t, + 12

is also an integral curve of X.

The proof of the corollary is obvious, but the fact that any two integral curves of X meeting at some point can be pieced together is an important idea to keep in mind when proving the global results in Theorem 5.33. In view of Theorems 5.31 and 5.32 there exists an integral curve at in for

X with maximal domain. We denote this integral curve by c,,, and, for emphasis, we mention once more that c' (t) = the domain of c. is an open interval containing 0, and we have cm(0) = in. The right- and left-hand end-points of the domain of c,,, are denoted by t+ (in) and t -(in), respectively.

Manifolds and vector bundles

135

Let 2x be the subset of M x P consisting of the points (m, t) such that C(m) < t < t+(nt)

and define the map F: 2x - M by F(nt, t) = cm(t). We call F the (global) flow of X. Note that 2x is the set of (m, t) E M x P such that X has an integral curve

c: I -+ M at m with t e 1. In particular 2x z) M x 0.

The vector field X is said to be complete if 2x = M x P. Thus, X is complete if and only if each integral curve can be extended so that its domain

becomes (- cc, oo); that is, t+(m) = oc and C (m) = - oo for all m e M. Examples (1) On M = P2, let X be the constant vector field, X(m) = (0, 1). The integral curve through m = (x, y) is t i-- (x, y + t). Each integral curve of X is defined for all t e P. so X is complete.

(2) On M = 082\0, let X be the same vector field. The integral curves are the same as before, except for the fact that any curve that passed through (0, 0) in Example (1) cannot now be extended to infinity; thus X is not complete. For example, t-(0, 1) = - I and t+(0, 1) = co.

(3) For M = Plet X E .x'(08) be defined by X(x) = _X2 . The integral curves of X are the solutions of do/dt = -[c(t)]2. Solving this differential equation

gives c(t) = (t + c(0)-')-' if c(0) # 0, and c(t) _- 0 if c(0) = 0. X is not complete; for example, t+(1) = cc but t-(1) = -1. Theorem 5.33 Let M be a manifold and X a £1(M). Then

(i) 1x is open in M x P and contains M x 0. (ii) The map F: !?x - M is smooth. (iii) If (m, t) E 2x then (F(m, t), s) E Zx f (m, t + s) a 2x; in this case F(m, t + s) = F(F(m, t), s)

Proof Let (m, t) E 2x. Both s i- cm(t + s) and s " cF,m,,j(s) are integral curves of X with the same initial condition at s = 0. Hence they must coincide and have the same (maximal) domain of definition. Taking s = t', (iii) follows immediately.

It remains to show that 2x is open in M x P and that F is smooth on 2x. It is clear from Theorem 5.31 that F is smooth in a neighbourhood of M x 0, but an additional argument is required for the other points of 2x. Let mo a M, and let J be the set of points t in the domain of c,,,0 such that F is C'° at (mo, t), that is, for which there exists a number b > 0 and a neighbourhood U of mo such that the product U x (t - b, t + b) is contained in 2x and the restriction of the flow F to this product is a C°° map. To show that 2x is open and F is C'° everywhere on 2x we must show that, for any mo e M, every point in the domain of belongs to J. We do this by a connectedness argument. It is clear that J is open. It remains to show that J is closed in the domain of c.0. Let s be in the closure

136

Boundary value problems for elliptic systems

of J; then by Theorem 5.31 there is a neighbourhood V of cma(s) and a number a > 0 such that the flow F is smooth on V x (-a, a).

(25)

Since c, is smooth (hence continuous) at s there exists t, e J such that Cmo(ti)

lies in V, c V. Since t, e J, the flow F is C' at (mo, t,). By the continuity of F there is a number b > 0 and neighbourhood U of mo such that

F maps the product U x (t, - b, t1 + b) into V

(26)

and F is C°° on this product. We can also suppose that It, - sJ < a. Thus for any in e U we have integral curves for X,

cm: (t, - b, t, + b) - M and

(-a, a) - M.

and, in view of the corollary to Theorem 5.32, every t e t1 + (-a, a) lies in the Cm(t) =

domain of the integral curve cm when in e U and we have t1). In other words, the product U x (t, - a, t, + a) is

contained in 2X and F(m, t) =

t 1), t - t 1)

for all (m, t) in this product. Hence in view of (25) and (26) the restriction

of F to this product, U x (t, - a, t, + a), is a composition of two C°° functions, so it is C°° itself. Since U x (t, - a, t, + a) is a neighbourhood of (mo, s) we have s e J. Thus J is both open and closed. By connectedness, J is the whole domain of Cma.

From now on we write the domain of the flow as 9 instead of 9X. Let 2, c M be the set of points in e M such that (in, t) e 9 = 9'r. Since -9 is open in M x R it is evident that 2, is open in M. The restriction of F to -9, defines a map F: 9, - M by F,(m) = F(m, t). Note that (iii) of Theorem 5.33 says that F, - F, = F:+, wherever it is defined. Corollary 5.34 Let X be a vector field on M and F: -9 - M its flow. Then .9, is open in M for each t e M, and F, is a dii feomorphism of.9, onto an open set in M. In fact, F,(9,) = 9_, and F,-1 = F_,.

Proof The fact that 1, is open has already been proved. Now let m e 9,. Then (m, t) e 2 and (iii) implies (F,(m), - t) e 1, since (in, 0) e 9 always holds. Hence F(m) e.9-, and F_,(F,(m)) = in

(inc 9,)

Similarly, if in e.9-, then F,(F_,(m)) = in.

Proposition 5.35 Let X e 5t'(M). Let c be an integral curve of X with a maximal domain of definition. If for every finite open interval (a, b) in its domain, c((a, b)) lies in a compact subset of M, then c is defined for all t e R.

Manifolds and vector bundles

137

Proof It suffices to show that the end-points a and b of the interval (a, b) lie in the domain of c. Let t e (a, b) with t. b. By passing to a subsequence converges to a point x e M. Now, since the domain we can assume that of the flow contains a neighbourhood of (x, 0), there is a neighbourhood U of x and a number r > 0 such that integral curves starting at points in U e U and persist for a time longer than T. If we choose n such that b - t < r, we can extend c to a time greater than b. Hence b is in the domain of c, since c has maximal domain. A similar argument holds for the other endpoint, a. The support of a vector field X defined on a manifold M is the closure of the set (m a M; X(m) # 0}.

Corollary 5.36 A vector field with compact support on a manifold M is complete. In particular, if M is a compact manifold then all vector fields on M are complete.

Note that when the vector field X is complete, the maps F, are diffeomorphisms M -+ M for each t e R and F, o F, = F+, for all s, t e R. The set (F; t e R) is called a one-parameter group of dii feomorphisms. Exercises 1.

Let c be a curve on M. Show that for all f e C°°(M)

2.

Let X e T(M), a vector field on M. Show that the directional derivative

X: C'(M) -+ C" (M) defined by Xf =df oX has the following properties

for f,geC°°(M)andi.aR:

X(f +g) = Xf + Xg, X a vector field on M, and let F denote its flow. Show that Xf = (d/dt)Fr(f )I,=0, that is, for all m e M, (Xf)(m) = d f(F,(m))I,=0

(f e C'O(M))

Let q: M -, N be a smooth map of manifolds. A vector field X e 1(M ) is said to be (p-related to Ye 1(N) if 4.

TipoX = Yore.

For smooth maps cp: M - N and 0: N --+ P show that if X is (p-related to Y and Y is cp-related to Z then X is (.y o q,)-related to Z.

Even when the vector field X e ((M) is not complete, the flows, F are local diffeomorphisms in the sense of the following definition (see [AMR]).

Boundary value problems for elliptic systems

138

Definition A flow box of X at m E M is a triple (U0, a, F) such that

(i) U0cMisopen,mEU0,andaeR,wherea>Oora=+ooandFisa smooth map UO x I. - M where I, = (-a, a). (ii) for each x e UO, the curve la - M given by t --+ F(x, t) is an integral curve

of X at the point x. (iii) if F,: UO - M is defined by F,(x) = F(x, t), then F,(Uo) is open, and F, is a diffeomorphism onto its image,

for all t e I". For each m e M, there exists a flow box of X at m, as the reader is asked to show in the next exercise. 5. Let X be a vector field on the manifold M. If the triple (U0, a, F) satisfies (i) and (ii) in the definition above, choose an open set Vo c UO and a positive

real number b < a such that F,(V0) c UO for all t c (- b, b). Show that (Yo, b, Flva,(-b,b)) is a flow box at m.

Let cp: M -+ N be a C' mapping, and let X e X(M) and Y e .E(N) be vector fields on M and N, with flows denoted FX and FT, respectively.

6.

(a) Show that X is q,-related to Y if and only if cp o F' = Fl o 4p. (b) If cp is a diffeomorphism, show that Y is the push-forward of X (i.e. Y = (p,X) if and only if the flow of Y is cp o FX o tp -'.

Hint for (a). For any curve c on M, we have ((p o c)'(t) = T(,)cp(c'(t)). 5.6 Partitions of unity

Before we can prove the existence of a partition of unity on a manifold M, we need the following lemma in R" which guarantees a plentiful supply of C°` functions.

Lemma 5.37 Let r, < r2. There exists a C°° function h: R" - [0, 1] such that h(x) = 1 if IxI < r,, 0 < h(x) < I if r, < IxI < r2 and h(x) = 0 if IxI > r2.

Proof By a scaling transformation we can assume r, = I and r2 = 3. The function cp: R -+ [0, 1] defined by

exp(-1/(1 - r2)),

Irl < 1

0,

Irl >, 1

(p(r) =

is easily shown to be C. Now let k

h(x) =kI

cp(r) dr > 0 and

2-IxI

(p(r)dr,

xeR".

Then h(x) equals 0 if IxI >, 3 and equals I if IxI < 1, and h is strictly between

0 and I if I < IxI < 3. Clearly h is C' when x 0 0. But h is constant near x = 0 so it is C' everywhere.

Manifolds and vector bundles

139

Corollary 5.38 Let M be a manifold and m e M. Then given any neighbourhood V of m there is a function A e C`°(M), 0 < A < 1, such that /i = 1 in a neighbourhood of m and k = 0 outside V. (We call A a "bump function".) Proof There exists a chart (U, K) at m such that U c V, K(m) = 0, and K(U) is the ball lxi < 3. By the lemma there exists a function h e C'(R"), 0 < h < 1, equal to 1 on lxi < 1 and vanishing when Jxi > 2. Let

A=

h-K on U 0

on CU

Since the support of h is a compact subset of K(U), it follows that A e C'(M).

Let M be a manifold. Recall that the support of a function f e C'(M) is the closure of the set of points x e M such that f(x) # 0. A C°° partition of unity on M consists of a system of smooth real-valued functions p j e C°°(M), j e J, satisfying the following conditions:

(a) for all x e M we have p,{x) > 0 (b) every point has a neighbourhood in which Ej pj is a finite sum (c) for each point x e M we have Y J pj(x) = 1 If { V, } jE j is an open covering of M we say that the partition of unity { p j } is

subordinate to { V J if supp p, e i' for all j e J. A collection T of subsets of a manifold M (or any topological space) is said to be locally finite if each point m e M has a neighbourhood U such that U n C = 0 except for finitely many C e W. Condition (b) says that the supports of p, are locally finite. It turns out that there exists a partition of unity subordinate to any given open cover. The open cover Y' = {V} is not assumed to be locally finite; however, the first step in the proof of existence of a partition of unity is to find a refinement of 'V which is locally finite. By a refinement of *- we mean another open cover {U;} such that for every i e 1, U, e V for some j. If the manifold M is compact one can just choose a flinite subcover, but in the non-compact case the following lemma is needed. Recall that a set in

a topological space is said to be a-compact if it is a countable union of compact sets.

Lemma 5.39 Any manifold M is a-compact. In fact, there exists a sequence of open sets G1, G2, ... such that

G" is compact, G" c G"+1, U G. = M. "=1

Proof In a countable basis for the topology of M, select those sets which have compact closure. The sequence U1, U2,. .. of open sets obtained in this way covers M, by local compactness of M. The G. Is can now be defined inductively as a union of these sets. Let G1 = U1, and suppose

G"= U, u U2u...uUj.

140

Boundary value problems for elliptic systems

has already been defined such that G" is compact for n -< N and G" a G"+1 for n < N. Since GN is compact it is covered by a finite number of the sets U,, U2. .... Then we let GN+, = U, j. u Uj,,, , where JN+ 1 is the smallest positive integer greater thanjN such that U, v v Uj.,,,, GN. The sequence of open sets G,, G2, ... defined in this way covers M, since j" - oo. A topological space is said to be paracompact if it is Hausdorff and every open cover has a locally finite refinement. Corollary 5.40 Any manifold is paracompact.

Proof Let G G2,... be a sequence of open sets with properties as in the lemma. Also let Gj = 0 when i <, 0. We have M = UR , A", where A" = G"\G"_, (intuitively, each A. is a closed "annulus"). Let { Vj } jE J be any open cover of M. Since { V, n (G"+1\6,-2)),,J is an open cover of A", and A. is compact, there is a finite set J. c J such that A. c U jeJ. Vj n (G"+ \6.- 2). In this way we obtain an open cover 1

{V n (G"+1\G"-2)}"EN.JE!

of M, which refines { V}1,, and is locally finite since a point in G, is not in G"+,\G"_ 2 when n >, 2 + i.

In the statement of the next theorem, B(0, r) denotes the open ball in I8" with center at the origin and radius r. Theorem 5.41 Let M be a manifold. Any open cover of M has a locally finite refinement consisting of a countable number of charts (Uj, rcj), i e N, such that i:j(Uj) = B(0, 3) and such that the open sets is '(B(0, 1)) cover M. There is a partition of unity { pj } subordinate to { Uj } with supp pi a compact subset of Uj.

Proof Let { V, } be a given open cover of M. For each x e M we can find an arbitrarily small chart (Ux, a,r) at x such that Kx(Ux) = B(0, 3). The proof of Corollary 5.40 shows that there is a countable number of charts (Ui, icj) such that { Uj } is a locally finite refinement of { Vj },

ic,(Uj) = B(0, 1) for each i, and the open sets W = is '(B(0, 1)) cover M.

Indeed, for each point x e A", we can find a chart (Ux, icx) at x such that icx(Ux) = B(0, 3), Ux c G"+1\G"_2 and U. c Vx for some jX. By compactness of A", finitely many of the sets W = icx'(B(0, 1)) cover A. We let {(Ui,'c )}

be the collection of charts obtained in this way for n = 1, 2, .. . By Lemma 5.37 there is a bump function h with h 3 0 everywhere, h = 1 on B(0, 1) and supp h e B(0, 3). Transporting h to the manifold

- h(Ki(x)), 0, {x)-j(

X e Uj

C0U,

we obtain a smooth function ipj e C'°(M ), 0 < 4/j <, I everywhere, such that

Manifolds and vector bundles

141

fir; = I in x;-'(B(0, 1)) and supp+0i, c U; is compact. Now we let Oi

and obtain a partition of unity subordinate to { U; } such that supp p, is a compact subset of U.

In general, an arbitrary open cover of M does not have a subordinate partition of unity consisting of functions with compact support. (For example, consider the real line R with the open cover consisting of precisely one open set, namely I8 itself.) However, we do have the following result. Corollary 5.42 Any open cover has a subordinate partition of unity.

Proof Let { V If be an open cover of M. There is a locally finite refinement, { U;), of { I') with the properties stated in the theorem, and a subordinate partition of unity {pi}. For each i, we have U; c V(;j for some j(i). We let (;)_ J; p;, where it is to be understood that (pj = 0 if j(i) -A j for all i. Then (p,_ypi=land

supp (p; c U

supp p, c V

for all j

ii: i(i) =i)

(For the first inclusion, recall that if {Ai} is a locally finite collection of closed sets then U Ai is closed). The supports of cp; are locally finite since any open set which intersects infinitely many of these supports would a priori intersect infinitely many of the supports of p;.

Corollary 5.43 Let U be open in M and let F be closed in M with F C U. Then there exists a C' function gyp: M supp qP C U.

[0, 1] with cp(x) = I for all x e F and

Proof By Corollary 5.42, the cover {U, M\F} has a partition of unity {ep, 1i} subordinate to it with supp rp c U and supp 0 e M\F. Then (p is the desired function. We will need the following in §5.11. It says that we can "shrink" a locally finite open cover of a manifold (this result is true for any paracompact space).

Lemma 5.44 If {U;} is a locally finite open cover of M, then there exists a locally finite open covering {I;} such that Vi a U,.

Proof Since M has a countable basis, any locally finite open covering is necessarily countable. So let us suppose that the cover {U;} is indexed by the natural numbers. Then

A,=M\(U.uU3u...) is a closed set contained in U,. Because M is paracompact, and hence a

142

Boundary value problems for elliptic systems

normal space, there is an open set V, with A, c V, c Y, c U,. Then

M= V, uU2.U3v and

is a closed set contained in U2. Let V2 be an open set with A2 c V2 c Y2 (-- U2.

Then we have M = V, v V2 u U3 v U4 v and continuing in this way it is not hard to establish by induction the following result: There exist open sets V,, V2, ... with i' c U, and such that for any j the open sets V V2, ... , V, U;+ Uj+2, ... cover M.

Now, for any x e M, there is a largest j with m e Uj, since { U,) is locally finite. It follows that x e V, u V2 'j because replacing U,+ 1. UJ+2, ... by J+1, V+2, ... cannot possibly eliminate x. Hence { l,} is an open cover of M and we are done. 5.7 Vector bundles

Let M be any manifold. Let { U, } be an open cover of M consisting of domains

of charts (U,, K;), and define the maps fi,: n-'(U,) -+ U, x I8" by

[c] i--' (x, dK;([c]x)) The pair (TM, it), where n: TM -+ M is the tangent bundle projection n((cjx) = x, satisfies the following properties: (i) Each fibre n-'(x) = TxM is a vector space. (ii) Each fi, is a difeomorphism, and the restriction to a fibre over any x e U; is a vector space isomorphism. This means that (TM, n) is a real vector bundle over M in the sense of the following definition. Definition 5.45 Let M be a manifold. A triple (E, p, M) where E is a manifold and p: E -+ M is a C'° map is called a real vector bundle over M if the following conditions are satisfied:

(i) for each x e M the fibre Ex = p-'(x) is a vector space over C$; (ii) there is an open cover {U,};E of M and, for each i e 1, a positive integer r and a d jeomorphism fi;: p-1(U;) - U; x ff8'

such that for all x e U;, #,(Ex) = {x} x R' and the map P;x: Ex -+ 08' induced by fi; on the fibres is a vector space isomorphism.

Complex vector bundles are defined in the same way if we replace R by C.

The maps /i, are called trivializing maps. Sometimes it is convenient to

assume that the U,'s are chart domains on M and then to extend the diffeomorphism f; by the coordinate map K;: U; - l; a R" to get coordinate maps y; _ (K; x id) o fi; for the bundle E, that is, Xi: p-1(U;)

where V. = K;(U;) c R".

id

U; x ff8 '

Y x !f8'

Manifolds and vector bundles

143

The number r may depend on i, but it is constant on each connected component of M. We will usually assume without mention that r is constant on all of M. (E, p, M) is then called a vector bundle of rank r. We can replace R' by an arbitrary r-dimensional vector space E, in which case we say that the vector bundle is modelled on E. This does not really enlarge the class of

vector bundles; any trivializing map f3: p''(U) - U x E can be composed with a linear isomorphism E - R' bringing us back to Definition 5.45. E is called the total space, and M is the base space. The map p is referred to as the vector bundle projection. Usually we refer to the total space E as the vector bundle, unless there is some reason to emphasize the role of the projection.

Let E, F be vector bundles over the same manifold M. A map f: E - F is said to be fibre-preserving if p'f = p, where p and p' denote the projections

of E and F on M, respectively. In other words, f is fibre-preserving if f(Ex) c F. for all x E M. In such a case we let f, denote the restriction of f to the fibre over x

ff:E.- FX and f is said to be "linear on each fibre" if fx is a linear map for each x. A C" map f: E -+ F that is fibre-preserving and linear on each fibre is called a homomorphism from E to F. The set of vector bundle homomorphisms from E to F is denoted HOM(E, F). A vector bundle isomorphism is a homomorphism f e HOM(E, F) for which there exists a homomorphism g e HOM(F, E) such that f og = idF and g , f = idE. If there exists an isomorphism from E to F we write E = F. The vector bundle (M x R', p, M), where p: M x68' -> M is the projection on the first factor, is referred to as a product bundle. More generally, a vector bundle E over M is called trivial if it is isomorphic to a product bundle, i.e.

E M x R' for some r. Lemma A map f: M x68' - M x R° is fibre-preserving and linear on each fibre if and only if it has the form f(x, v) = (x, Ax) 1)

for some map f: M -+ L(R', Rd). If

(`)

holds then f is smooth if and only if

is smooth.

f

Proof The first statement is obvious. For the second statement, the "if" part is clear. Conversely, suppose that f is smooth. Let pr denote the projection

M x Rd .+ Rd on the second factor. This is a linear map, hence C. Let e...... e, be the standard basis of R'. Taking v = e; we find that f(x)e, = pr ° f(x, ei)

is smooth in x. But f(x)e4 is just the ith column of the matrix f(x), Remark As usual we identify L(R', Rd) with the space of d x r real matrices.

144

Boundary value problems for elliptic systems

If f e HOM(E, F) and (U, fi) and (U, r/') are local trivializations of E and F, respectively, then, since f is fibre-preserving, the map

f",s=0°f°f-1:U x l'

U X Rd

(d = dimension of the fibres of F) has the form (x, v) i - (x, AX) v),

where f: U -+ L(R', Rd) is smooth. We refer to f as the local representation of f with respect to given trivializing maps. The following properties hold: (i) The local representation of the identity map f = idE e HOM(E, E) is the r x r identity matrix; (ii) if f c- HOM(E, F) and g c- HOM(F, G) and (U, Ii), (U, 0) and (U, ip) are local trivializations of E, F and G, respectively, then (g o f ).p = g.,,. c This means that the local representation of g o f is the map x i - g-(X) f (x) (matrix multiplication).

Lemma 5.46 Let E, F be vector bundles over M. If f e HOM(E, F) and is bijective, then f is a vector bundle isomorphism.

Proof Since f is fibre-preserving we have f(Ex) c Fx for all x c- M. But f is

also bijective so it must be that f(Ex) = Fx. The set-theoretic inverse g: F -+ E is therefore fibre-preserving and it is linear on each fibre since f is. To prove that g is smooth we can work locally. The local representation of f has the form f: U -. GL(R'), and then the local representation of g is the map U -+ GL(fE') given by g(x) = [ f(x)] '. In the notation of Lemma 5.2 this means that g = J o f, so g is C°° and hence g is C'. Let E be a vector bundle over M, and choose an open cover { U; },E l of M and trivializing maps /3, as in Definition 5.45. The maps h,;: U; n UI GL(f8') defined by

hy(x) = (I 1 1)x are called the transition matrices (or functions) for E, and we have h!, =

(hi,)-1

in L

U,

in U,nUjnU,.

hit' hjk = hik

these properties being known as the cocycle conditions. Given a system (h,,)

of r x r matrix functions with C' entries satisfying the cocycle conditions, one can define a vector bundle having hip as its transition matrices by forming

the disjoint union of all U, x f8', i e 1,

Q=U{i}xU,xli', iel

and by identifying

(i,X,v)EU, X f8'-(j,x,v)eU, x l' if v' = h f,(x) - v. It follows from the cocycle conditions that this is an equivalence relation - on Q, and it follows from Proposition 5.47 below

Manifolds and vector bundles

145

that the quotient set S = Q/- is a vector bundle with projection p: S M defined by [(i, x, v)] i--' x and trivialization f3,: p-1(U,) -' U; x68' given by [(i, x, v)] r+ (x, v). Also (S, p, M) is isomorphic to E if h,, were obtained from local trivializations of E as explained above.

The following proposition states the axioms by which a collection of trivializing maps on an arbitrary set S define a vector bundle structure on S.

Proposition 5.47 Let M be a manifold, and p: S - M a mapping from some set S into M. Let {U1} be an open cover of M and, for each i, suppose we are given a bijection /3,: p-1(U,) -+ U, x 68' satisfying the following properties: VB I The map /3, is fibre-preserving, i.e. the diagram

p-1(U,)

-,,

U, x 68'

.+ /f

U,

commutes. Consequently, there is an induced map on the fibres, which we denote by fl, : p-'(x) - 1W. VB 2

If U, and U, are two members of the cover such that U, n U, 0 0, then for each x a U, n U,, the map (/3, f; 1)x a L(68') is a linear isomorphism, and the map of U, n U, into GL(R') given by

is C'. Then there is a unique topology and C' structure on S, and a unique vector space structure on each fibre p''(x), such that fl,., is a linear isomorphism and (S, p, M) is a vector bundle with /3, as trivializing maps.

Proof For x e U; n U;, let h,;(x) = (f, By hypothesis, h;; is a smooth map U, n U; - GL(68'). It follows that the overlap maps

(U;nU;)x68'->(U,nU;)x1W are given by (x, v) - (x, h,,4x)v) and are therefore smooth. These maps are in fact diffeomorphisms since hi;' = h;, is also smooth. We claim that by Proposition 5.16 there is a unique C' structure on S such that the #i's are diffeomorphisms. The collection {(p-'(U,), /3;)) is not quite an atlas for S since the image of 1, is not a subset of Euclidean space. But we can argue as follows: each U, is a union of chart domains so we may assume without loss of generality that we are given an open cover (U,) with charts (U;, K,) for M. We then let Xi = (ic, x id) o /3; to obtain a bijection X;: p -1(Ui) - K,(U;) x 68' c 68" x 68'

which satisfies AT 1, AT 2 and AT 3 of Proposition 5.16. Hence there is a

unique C' structure on S such that the X,'s (and hence the /t; s) are diffeomorphisms.

It is clear that p is smooth, since p: S - M is locally obtained as a composite of smooth maps, i.e., pI e, = p; ° P; where pi: U; x 18' - U; is the

146

Boundary value problems for elliptic systems

projection on the first factor. If x e U, we can transport the vector space structure of R' to the fibre p-'(x) by means of fi,,,. The result is independent of the choice of Ui since (Yi fi! '),, is a linear isomorphism.

Remark When applying Proposition 5.47 it often happens that the fibres p'(x) already possess a vector space structure such that fix is linear. The vector space structure defined in the proof will agree with the given structure.

Let (E, p, M) be a vector bundle over M. A smooth map v: M -. E is called a section of E if p o v is the identity on M; in other words,

v(x) a E., = p-'(x)

for all x e M.

The space of all sections of E is denoted C'°(M, E). If U is an open set in M we refer to a section s e C°°(U, El v) as a section of E over U.

Let u e C'(M, E) be a section of E. If we have a covering {U,} of M as above with local trivializations /3,: EIv, - U, x R', then ui = u e C I(U1, R') and we have

in U, n Uj,

u, = hi j ut

(27)

where h,1 are the transition matrices introduced above. Conversely, any system u, a C'°(Ui, R') satisfying (27) defines a section of the vector bundle. Lemma 5.48 A collection of sections v1,. .. , v, a C°°(U, E) is called a local basis for E if for every point x in U, the vectors v,(x), ... , v,(x) form a basis in the vector space E, E has a local basis over U if and only if EJ , is trivial.

Proof If fi: EIv -+ U x R' is a trivializing map and e, denote the standard unit vectors in R' then the sections i = 1, ... , r

vi(x) = (Q:)-1(ei),

are a local basis on U. Conversely, if vi, ... , v, is a local basis on U, then the map ip from U x R' to Elv defined by (x,

i=1

(28)

Vi(x)

is a vector bundle isomorphism (it has a smooth inverse by Lemma 5.46). Hence fi = cp -' is a trivializing map. The following corollary is the analogue of Proposition 5.29. Corollary 5.49 Let E be a vector bundle over M, and let v1,. .. , v, be a local basis on U. If v is a section of E over U, that is, v e CI(U, E) then we have r

f(X)-VAX),

v(X) _

xeU

i=1

f o r unique functions f e C'"(U), i = 1, ... , r. Conversely, any map v: U -' E defined in this way is a section of E over U.

Manifolds and vector bundles

147

Proof Let fl: El - U x R' denote the trivializing map defined as the inverse of the map (28). Then

s\,-t L Si

l\`

ss

zz

V1(x))=(x,S1,...,Sr)

XEU,i;ER'

Now, /3 v is a section of the trivial bundle U x R', so we have

fi(v(x)) = (x, P0 for a unique function f e C' (U, R'). This proves the first statement since f (x) = Ei =1 fi(x) - e; and vi(x) = (/x) -' (e; ). The second statement is obvious.

If E = M x R' is a trivial bundle the space of sections of E can be identified with the smooth vector-valued maps M - R' and we write C'(M, R') rather than C'°(M, E). If r = 1 we write C'(M). Note that E is trivial if and only if there exists a global basis, that is, a basis defined on all M. Examples

(1) Let S' denote the unit circle lxi = 1 in R2. Since S' is a submanifold of R2 the tangent space to S' at any point is identified with a subspace of R2; at the point x = (x,, x2), we have

T.(S')_ {I;ER2;x1s, +x2 2 =0} We say that RS') is a line bundle because its fibers are one-dimensional. Note that T(S') has a non-vanishing section: the section v: S' - T(S') defined by (X1, x2) i-' (-x2, x1). Hence T(S') is trivial because there is the vector bundle isomorphism S' x R - T(S 1) defined by (x, A) i- A v(x).

(2) Let T" = S' x . . . x S' (n copies) be the n-torus. Then

71S') x ... x T(S') and, since T(S') = S' x R, we have T(T") = T" x R". Thus the tangent 7IT")

bundle to the n-torus is trivial. (3) Let S2 denote the unit sphere lxi = 1 in R3. At any point x = (x1, x2, x3) we have T;(S2) = {S E R3; x161 + x22 + x3c3 = 0)

Using degree theory it is possible to prove that T(S2) does not have a non-vanishing section (see [AMR], §7.5). This implies that T(S2) is non-trivial.

(4) (Mobius band) Next we give an example of a non-trivial vector bundle

dl over S'. On the product manifold R x R, consider the equivalence relation defined by

(t, v)-(t',v')-t'=t+ k,v'=(-1)kv

for some keZ and denote by n: R x R - !l the quotient topological space...!! is called

Manifolds and vector bundles

149

end points must match under the twist, and hence s must cross the 0 section at some point. Now we verify this fact analytically. Suppose that there is a non-vanishing section s: S1 -* Al. Let c be the closed curve in Al defined by c(t) = s(e2n"), 0 < t 5 1. Recall that 3, defines a trivialization of l(I v, and let f: (0, 1) -+ R be defined by f(t) = second component of fl,(c(t)).

A simple argument shows that f can be continuously extended to [0, 1]. In fact, near t = 0, f(t) is equal to the second component of /32(c(t)), and near t = 1 it is equal to minus the second component of f2(c(t)). Now let c: [0, l] R x R be defined by c(t) = (t, f(t)). Then c is a "lifting" of c in the sense that c = n o c, that is,

c(t)=[t,f(t)],0,t,1 f(0). The Since c(0) = c(l) we find that [0, f(0)] = [1, f(l)]. Hence f(l) continuity of f and the fact that [0, 1] is connected implies that f(to) = 0 for some to. Then c(to) = [to, 0] = 0, which contradicts the assumption that s is non-vanishing. Sub-bundles

Let (E, p, M) be a vector bundle over M. A subset F c E is called a sub-bundle if for each x e M there is an open set U containing x and a local trivialization fl: p-1(U) -+ U x R' such that SB

/3(p-1(U) n F) = U x (08'-k x 0).

Proposition 5.50 Let (E, p, M) be a vector bundle over M. If F is a sub-bundle of E, then (F, pIF, M) is a vector bundle over M such that the inclusion map F -+ E is a vector bundle homomorphism.

Proof Let 1, Ui} be an open cover of M such that there exist local trivializations /3i: p- 1(U;) -+ U; x 11' satisfying the SB property relative to F. The conditions VB 1 and VB 2 of Proposition 5.47 are easily verified so (F, pI F, M)

has a vector bundle structure. For instance, note that if hi,(x) _ (/1,

are the transition functions for E. then F has transition functions h;,{x) = I1,,01 R..11 ,,1 which are smooth from U, r Uf to GL(f3'-"). The inclusion map F -' E is obviously smooth because if /3 is a trivializing map satisfying the SB property then its local representation with respect to

#IF and $ is the inclusion U x (R''* x 0) - U x R r. Sub-bundles have the following universal property. Proposition 5.51 Let F be a sub-bundle of E, and G another vector bundle over M. Let f: G -+ E be a vector bundle homomorphism such that f(G) e F. Then the induced map f,: G - F is a vector bundle homomorphism.

Boundary value problems for elliptic systems

150

The proof is similar to Proposition 5.23. Also note that the vector bundle structure on F is uniquely determined by the requirement that the inclusion map F -+ E be a vector bundle homomorphism.

Proposition 552 Let E and F be vector bundles over M. If f: E -+ F is a vector bundle homomorphism let f,: Ex -+ Fx be the restriction off to the fibre over x e M, and define the kernel and image off by

ker(f) = U ker(f)

and

im(f) = U im(fx)

xEM

XEM

If the rank of fx is locally constant then ker(f) is a sub-bundle of E and im(f ) is a sub-bundle of F.

Proof Let U be an open set in M such that there exist trivializing maps (3: Ev -+ U x R' and ki: Fu - U lFl". The local representation

ft,=0°.f °$-':U fl-: X68' -+U X Rd has the form (x, v) -+ (x, f(x)v), where f: U L(Il ', II") is smooth. Fix xo e U. If we let R' = 'V1 ®Y'2 where I' is the kernel of f (xo) and Rd = -YVI (j)'# z where 7f is the image of f (xo), then f (x) has the block matrix representation

f(x) =

[aii(x) a21(x)

a12(x)1

E L(Y'l 0 12,

1 (D Y12)

a22(x)

where aii: U -+ L(Yr, Wi) is smooth. The choice of Y'; and W' implies that a12(xo) is an isomorphism. Therefore, a12(x) is an isomorphism for all x in a neighbourhood of x0; by shrinking U we can assume that this holds for all

x e U. Note also that a11(xo) = 0, a21(xo) = 0 and a22(xo) = 0. Since the rank of f(x) is locally constant we can shrink U further, if necessary, so that f(x) has constant rank for all x e U. Since aI 2(x) is an isomorphism and the rank of f(x) is equal to dim YW, it follows that a2t(x) = 0 and a22(X)=O for all x E U Now if we let A(x)

=[

I - a12(x)-''a11(x)

0

I

C_ GL(Y e I-, Y- D I-)

then

1(Y) A(r) =

C0

a,2(x)

0

0

(29)

Replacing the trivializing map P by A-'#, we may assume that f(x) is equal to the matrix on the right-hand side of (29). Then we have ker f(x) = Y', and im f(x) = Y!; for all x e U. Finally, by a linear isomorphism (change of basis) in 68' and I we can assume that Y, = R'-k x 0 and 'Y! i = 08" x 0, where k is the rank of f(x). Now we have (fl'"k x 0) and ifi(FF n im(f )) = U x (Ilk x 0) fl(Ev n ker(f )) = U x which is the SB property for ker(f) and im(f).

Manifolds and vector bundles

151

Remark From the proof of the proposition we see that if f: E F is any homomorphism then rank(fx) rank(ff0) for all x near x0. Thus, rank(f) is an upper semi-continuous function of x. Definition 553 A projection operator P for a vector bundle E is a homomorphism P: E - E with p2 = P. If P is a projection operator for E, then im(P,,) a im(I - P.) = E., and

rank(Px) + rank(I - P,) = dim E. Since both rank(P) and rank(I - P.) are upper semi-continuous functions of x, they are locally constant. Thus by Proposition 5.52 both ker(P) = im(1 - P) and im(P) are sub-bundles of E and E = im(P) ® im(1 - P). Proposition 554 Let (E, p, M) be a vector bundle over M, and F e E a sub-bundle. Let EJF be the (disjoint) union of the quotient vector spaces E/F,, x e M. Then ELF has a unique vector bundle structure over M for which the canonical map n: E ELF is a vector bundle homomorphism.

Proof Let P: E/F --* M be the map induced from p: E - M. By the SB property, there is an open cover { U; } of M and local trivializations Jar: Eau, -+ Ur x R' such that /J1(Flu,) = U, x (R'-" x 0). Then l; induces a unique map

A:P-'(Ui)

Uix(0xRk)

by the condition fir o n = AI0 1(v, _ (o Rk) . Note that /3, is a bijection and property VB 1 holds. To verify VB 2, let h denote the transition function x (-+ (fir c / ')X for E. We claim that the transition function h for E/F given by x F- (/3, /j 1)s is also smooth. Writing h as a block matrix with respect (Rr-k to the direct sum 68' = x 0) ® (0 x Rk) we have h(x)

h12(x)1

= h2,(x)

h22(x)

where h,1, h12, h2, and h22 are smooth matrix functions on Ur n U, of dimensions (r - k) x (r - k), (r - k) x k, k x (r - k) and k x k, respectively. Since 18'-k x 0 has to be carried into itself by h, we have h21 = 0. Since h(x) is invertible, then so is h22(x) for each x. The proof is complete since h = h22. By Proposition 5.47, (E/F, P, M) is a vector bundle. It is evident from the

local representations that the canonical map it is a vector bundle homomorphism (i.e. smooth). Exercises 1.

Let Y c M be a submanifold.

(a) Show that TY is a sub-bundle of TMI y.

Boundary value problems for elliptic systems

152

(b) The

bundle of Y is defined to be the quotient bundle v( Y) _

(TMI).)/TY. Let k denote the codimension of Y. Show that v(Y) is trivial if there are smooth maps X;: Y - , TM, i = 1, ... , k, such that X;(y) a T,.M and X, (y), ... , Xk(y) span a subspace Vr satisfying T,. M = T. Y ® 1 . for all y e Y. 2.

Let S" c It"+' be the sphere. Show that the normal bundle of S" is trivial.

Let E be a finite dimensional vector space and let F c E be a subspace. The annihilator of F, denoted F°, is the set of i. E E* such that <1., v> = 0

for all v e F.

There is a natural isomorphism F° -_ (E/F)*. 3. Let Y c M be a submanifold. The conormal bundle, p(Y), is defined to be the union of the annihilators (T,. Y)° c Ts M, y e Y, i.e.

p(Y)={SeT,*.M;<S,v>=0forall veT.Y, yc- Y} (a) Show that p(Y) is a sub-bundle of T*MI,.. (b) Show that p(Y) = v(Y)*, i.e. the conormal bundle is isomorphic to the dual of the normal bundle. 5.8 Operations on vector bundles

Let Y be a submanifold of M. If E is a vector bundle on M we can restrict it to Y

Ely= U rE,.

(=p-(Y))

For the sake of simplifying the notation we also write Ey = Ely. Each fibre of Ey is a vector space (being a fibre of E) and we would like to show that it is a vector bundle. The crucial step is to show first that Ey is a submanifold of E.

Now, for each y e Y there exists an open set U containing y and a local trivialization /3: p-'(U) -+ U x R'. By shrinking U we may assume that it is the domain of a coordinate map A satisfying the submanifold property SM:

k(U n Y) = K(U) n (IF"-k x 0)

(k = codimension of Yin M)

Then the composite map y = (K x id) /1, that is, y: p

-'(U) B

U x 98'

x(U) x Q8' c G8"'',

satisfies the following condition:

y.(p-'(U) n Ey) = !.(p-'(U)) . (R"-' x 0 x R') Thus (p-'(U), X) is a chart on E that satisfies the SM property for E,. in E; this shows that Ey is a submanifold (also of codimension k) in E.

Manifolds and vector bundles

153

Proposition 5.55 Let E be a vector bundle with projection p: E -+ M, and let Y be a submanifold of M. Then

Er=UEy yer

is a submanifold of E and a vector bundle on Y with projection ple,: Er - Y.

Proof We have already shown that Er is a submanifold of E. Now, plE, is a C°° map since it is the restriction of a C'° map to a submanifold (see Corollary 5.23). For the same reason, if we have a collection of local trivializations of E,

p-'(U;)

U, x R r'

such that { Ui } cover M, and restrict them to E, we obtain a collection of local trivializations of Er,

p- '(U;) n Er -' (U; n Y) x such that the sets UJ n Y cover Y. (The restriction is C'° and the same holds

for the restriction of the inverse since (U; n Y) x R' is a submanifold of U;x1W.)

Let N and M be manifolds and (E, p, M) a vector bundle over M. Any

smooth map f : N - M induces a vector bundle f '' E on N, called the pull-back of E by/. The pull-back / - 'E is the subset of N x E consisting of pairs (n, e) e N x E with p(e) = f (n); it is the unique maximal subset of N x E that makes the following diagram commutative:

-

Ipl

Iv

NfM

where the vertical map / `p on the left is given by (n, e) 1-4 n and the top horizontal map is (n, e) F-+ e. The fibre off `E over a point n e N is

V-'E) = {n} x which we identify with E,,(.,, and thus has a vector space structure. We also give/ - 'E the relative topology as a subset of N x E. Example Suppose that E = M x 1W, and p the projection on the first factor.

Then if f: N - M we have

f -'(M x R')= {(n, f(n),v);neN,ve1W) =

r x 118'

where r is the graph of f. By virtue of Exercise 5 in §5.4, f ''(M x 1W) is a submanifold of N x (M x 1W).

154

Boundary value problems for elliptic systems

Suppose E is trivial, that is, there is an isomorphism rp: M x68' -+ E. Then

the diffeomorphism idN x rp: N x (M x68') -' N x E maps f `(M x P') to/ -'(E). Hence/ - `(E) is a submanifold of N x E (see §5.4, Exercise 1(b)).

Since any vector bundle is locally trivial, it follows that / `E is a submanifold of N x E when E is any vector bundle over M. Indeed, let U be an open set in M such that Ev is trivial. Then V =f -'(U) is open in N, and the pull-back of Ev under the map /I v: V --+ U is a submanifold of V x Ev. Note also that this pull-back is equal to

f-'EIv=(VxEu)nf-'E which is open in/ - `E since V x Ev is open in N x E. Thus, every point of

/- ' E belongs to an open set (in f `E) which is a submanifold of N x E, and, consequently,/ -'E is itself a submanifold. Proposition 5.56 Let f : N -. M be a smooth map and E a vector bundle with projection p: E - M. Then f -'E is a submanifold of N x E, and is a vector

bundle over N with projection f `p: f - 'E - N. Proof In fact, if { U; } is an open cover of M such that Ii,: EIv, - U, x 68' are trivializing maps, let A2 denote the second component of f, (the composition of f, with the map Ui x 68' 6t'). Then the sets V =/ -'(U,) are open and cover N, and the maps (p,: f -'EIv, - V xpp 68' (n, e) f-' (n, Nt2(e))

are trivializing maps for/ -'E. Each map 4p, is C'° since it is the restriction of a smooth map on N x E to a submanifold of N x E. The inverse of 4p, is also smooth due to Lemma 5.46. Natural operations on vector spaces carry over to vector bundles. In this section it is convenient to extend slightly the generality of Proposition 5.47. We can replace R' by any r-dimensional vector space E. The bijections fl,: p- `(U,) -+ U, x E satisfying VB I and VB 2 then define a vector bundle structure on S modelled on E. The transition functions h,j are smooth maps

U, n U - GL(E). As the first example of an operation on vector bundles, we consider the direct sum. If E and F are vector bundles over M modelled on vector spaces E and F, then

E®F= U Ex®F has a vector bundle structure over M modelled on E (D F. In fact, there exists an open cover {U,} of M and trivializing maps

F -+ U x F

(30)

Manifolds and vector bundles

155

so that we have a bijection

f#r:(E®flu, - U, x (E(D F) and, for each x e M, fJr: = cpu ® cp,X is a linear map E. ®Fx -, E ®F. Thus VB I and the first part of VB 2 are satisfied; the map p: E ® F -, M

is the obvious one defined by p(x) = E. ®F. for each x e M. As for requirement, let hij(x) = (q', q ')x a GL(E) and c- GL(F) be the transition functions for E and F, respectively. Then E (D F has transition functions U, n UU -+ GL(E ® F) given by the

smoothness ((p,

0

0

J

which are smooth since h,j and h,, are smooth. By Proposition 5.47, E ® F is a vector bundle modelled on E ® F. If E is rank r and F is rank d then E ® F is rank r + d (since R'(D Rd -_ R" 1d). Similarly, we can define other natural operations on vector bundles. For the tensor product E ® F = U E® ® F, given the trivializing maps (30) we have the bijection

Pr: (E ®F)v, - Ur x (E (D F),

(31)

and, for each x e M, it is a linear map E® ® F,, -+ E 0 F defined by A. = (pi. (& (p,.. The transition functions x r-+ hr,.{x) 0 ht(x) E GL(E (& F) are

again smooth, so by Proposition 5.47, E ® F is a vector bundle modelled on E 0 F. If E has rank r and F has rank d then E ® F has rank (since R' (9 Rd - R' d). For the dual bundle E* = U (Ex)*, the transition functions are not quite

so immediate. Recall that for a real vector space E, the dual space is E* = L(E, R), the real-valued linear functionals on E. A linear map f: E -+ F

induces the transpose map f: F* - E* defined by

(f(a),v)_(a,f(v)>,

aeF*,veE,

that is, f(a) = a- f (a c- F*). Note that the transpose operator t is a linear map L(E, F) -+ L(F*, E*). Now, if 0j: Eu6 -+ Ur x E' is a trivializing map for E, then

fir (E*)v, - U, x E* where fiu is a bijection which has transition functions x r-+ ($, Q; 1). = [`h, (x)] ` a GL(E* ) where h,Ax) = (pi ' )x E GL(E). Since the transpose operator t is linear, hence smooth, these transition functions are smooth in view of Proposition 5.2. Thus E* is a vector bundle over M modelled on E*. If E is rank r then

E* is also rank r since (R')* W. These types of constructions of vector bundles can be unified with the language of category theory. Proposition 5.57 and the series of exercises following it are meant to illustrate this idea.

Boundary value problems for elliptic systems

156

Recall that a category 21 consists of a class of objects and for any two objects A and B, a set Hom(A, B) of morphisms from A to B, satisfying the

following properties. If f is a morphism from A to B and g a morphism from B to C, then the composite morphism g o f from A to C is defined; furthermore, the composition operation is required to be associative and to have an identity 1A in Hom(A, A) for every object A. Some examples of categories are:

The class of all topological spaces, with morphisms being the continuous maps. The class of finite dimensional vector spaces, with morphisms being the linear maps. The class of all groups, with morphisms being the group homomorphisms.

For a concise introduction to categories, see [Do]. A covariant functor T from a category 21 to a category S is a rule which to each object A in 91 associates an object T(A) in 23 and with each morphism

f: A

B a morphism t(f): r(A) - r(B) such that

r(f ° g) = r(f) ° r(g) whenever f and g are morphisms such that f o g is defined, and t(14) = 1,(4)

(32)

(33)

A contravariant functor r is defined by reversing the arrows in (32) in the following sense: t is a rule which to each object A in 21 associates an object t(A) in 2i and with each morphism f: A -+ B a morphism t(f ): r(B) - T(A) (going in the opposite direction) such that (33) holds and (32') t(f ° g) = t(g) ° T(f) whenever f and g are morphisms such that f c g is defined. From now on, we always work with the category 21 of finite dimensional vector spaces over O k with morphisms being the linear maps. Let t be a covariant functor on W. Then the functor r defines a map

r: L(E, F) - L(t(E), T(F)); we say that T is a smooth functor if this map is smooth for any real finite dimensional vector spaces E and F. Let VB(M, 21) denote the category of vector bundles over M modelled on vector spaces in IN, with morphisms taken to be the vector bundle homomorphisms. The next proposition shows that a smooth functor on 21 can be extended to VB(M, 21). Proposition 5.57 Suppose that r is a smooth covariant functor on 91, the category of finite dimensional real vector spaces. If E is a vector bundle over M, define the family of vector spaces,

r(E) = U T(Ex) x

(disjoint union)

Manifolds and vector bundles

157

r(F) F is a vector bundle homomorphism, define the map T(f ): r(E) such that its value on the fibre over x is given by r(fS): T(E,,) -+ T(FS). Then, for each E e VB(M, RI), there is a topology and C' structure on T(E) 1f f: E

such that r(E) e VB(M, RI) and the following properties hold: (a) If f : E -+ F is a vector bundle homomorphism then T(f) is a vector bundle homomorphism T(E) -+ T(F). (b) r is a functor: r(f a g) = r(f) o r(g) whenever fog is defined, and r(idL) =

(c) If E = M x E is a trivial vector bundle then T(E) = M x r(E) is a trivial vector bundle.

(d) 1f f : N -+ M is a smooth map between manifolds then r(f -1E) = f 'r(E).

(To be precise in (d) we should write T,(j -'E) =/ -'TM(E); that is, there is a distinct functor r = TM for each manifold M. But (d) shows that there is no harm in dropping the reference to the manifold.)

Proof As a preliminary remark, note that it follows from (32), (33) that if

f: E -+ F is an isomorphism then so is r(f): r(E) - r(F), and in fact

T(f)-' =T(f-').

We assume that M is connected, so that each fibre is isomorphic to a fixed vector space E. Let { U; } be an open cover of M with trivializing maps for E

/J:EIv,-+U,xE Then we have a bijection

T(fi): T(Elv,) - U, x r(E) defined on the fibers over x e U, by the linear isomorphism r(f ix): T(E)

We must show that these bijections satisfy the conditions of Proposition 5.47. There is an obvious map a: r(E) -+ M such that VB I holds for (r(E), n, M). To verify VB 2 note that the transition functions for r(E) are x F-- r(fi ) a r(f%x)- 1

The expression on the right is equal to r((/li o fJj '

T(h,t(x)) where hip are

the transition functions for E. Since h,; is smooth from U, n Uj to GL(E) then r a h,j is smooth from U; n UU to GL(L(E)) due to the smoothness property of the functor r. Thus there is a unique topology and C°' structure on r(E) such that r(fl,,,) is a linear isomorphism on each fibre n-'(x) = r(E) it, M) is a vector bundle with r(fi) as trivializing maps. Now, for each vector bundle homomorphism f: E - F we have the map r(f ): r(E) -+ T(F) which is obviously a vector bundle homomorphism, provided we can show that it is C'. To do so we can argue locally. We have the commutative diagram

EIv, - Flv, 1e,

14,

U,xE -i UixIF where the map in the bottom row is (x, v) i- (x, f(x)v), and by the functorial

Boundary value problems for elliptic systems

158

property of r we obtain another commutative diagram T(f)

T(Elv)

r(F'Iv,)

U; x r(E)

Ui x r(F)

where the map in the bottom row is (x, w) f- (x, r(f(x))w). The map f: U; -+ L(E, F) is the local representation of f_and is smooth, so that by the smoothness property of r, the map x -. r(f(x)) is also smooth. Hence r(f) is smooth. Conditions (b) and (c) are obviously satisfied. We leave the proof of (d) as an exercise.

Examples (1) Let r(E) = nk(E), the kth exterior product (see §5.1). If f e L(E, F) there

is a unique linear map r(f) = nkf e L(AE, AF) such that A ... A f(vk)

V1 A ... A

(34)

Then r is a covariant functor on the category of finite dimensional real vector spaces, that is, A(f - g) = nkf c nkg whenever f - g is defined and Aide) = idME. (2) Let r(E) = E* and for each linear map f e L(E, F) define the linear map r(f) = f e L(F*, E*). Then r is a contravariant functor on the category of finite dimensional real vector spaces. One can also define functors of several variables r(E, F,.. .... ), covariant in some variables, contravariant in others. For instance, a functor r in two variables is said to be contravariant in the first variable and covariant in the second if r(fi `f2, g) = r(f2, g) ° r(f1, g) r(f, g1 "g2) = r(f, g1) ° r(f, g2) and r(idE, idF) = id,IE,FI. Further, r is said to be smooth if the map

L(E', E) x L(F, F') - L(r(E, F), r(E', F')) defined by (f, g)"r(f, g) is smooth. See Exercise 7 below. Exercises

Let Y be a submanifold of M, and e': Y - M the inclusion map. Show that 1E = El r. Hint: The isomorphism is given by (y, e) i- a where y e Y, 1.

e e E, with inverse e F- (p(e), e). 2.

V

(a)

Let f : N -, M and y: Z - N be smooth maps. Show that

°p)-'E

1VIE)

(b) Show that isomorphic vector bundles have isomorphic pull-backs.

Manifolds and vector bundles 3.

159

Show that nk is a smooth covariant functor in the category 21 of finite

dimensional real vector spaces. That is, if f e L(E, F) then there is the induced linear map nkf a L(nkE, nkF) defined by V1 n ... A Vk~f(V1) A ... A f(VL)

(34)

and it has the following properties:

(i) If g e L(F, G) and f e L(E, F) then nk(g f) = nkg, nk (ii) If idE a L(E, E) denotes the identity on E then A (idE) = idnkE, and the smoothness property:

f

(iii) The map nk: L(E, F) -' L(nkE, nkF) is C'. Hint for (iii). If bases are selected for the vector spaces, then / \ becomes a matrix function with polynomial entries.

Let r be a smooth covariant functor on 21. Let E be a vector bundle over M, modelled on E, and let SE C''(M, t(E)) be a section, that is,

a smooth map s: M -, t(E) such that s(x) E r(Ex) for all x e M. If /i: EI,; -' U x E is a trivializing map then we obtain an isomorphism

E, hence an isomorphism t(/3.,): r(E.,) -+ t(E) for each x. Thus, s has a local representation r(E) (') is U where i is smooth. /3x: E.

4.

Consider once again the functor N. Write out in detail what the local

representation (*) of a section s e C'(M, Ak(E)) looks like in this case, given

a basis v...... v, in E. Show that Proposition 5.57 holds for smooth contravariant functors, where in (a) the arrow is reversed. (A contravariant functor gives a map L(E, F) - L(t(F), t(E)) and, as before, t is said to be a smooth functor if this map is smooth for all E and F.) 5.

and for each f e L(E, F) let r(J) = f e E") be the Let T(E) = transpose of f, i.e. t(f )h = h - f. Show that t is a smooth contravariant 6.

functor on the category of finite dimensional real vector spaces.

The functors considered in Proposition 5.57 and Exercises 3 to 6 are functors of a single variable. Similarly, one can define functors of many variables, contravariant in some variables and covariant in others. 7.

Let t be a smooth functor of two variables in the category 21,

contravariant in the first variable and covariant in the second. If E and F are vector bundles over M, define

T(E,F)=Ut(EY,F.), s

and, for each pair of vector bundle homomorphisms f: E' -, E and

Boundary value problems for elliptic systems

160

V. define the map

g: F

t(f, g): t(E, F) -+ r(E', F') whose value on the fibres over x is given by t(fx, gx): t(E, F.) - t(Ex, F."). Show that t is a functor in the category of vector bundles with properties analogous to (a)-(d) of Proposition 5.57. 8.

Let Hom or L be the functor which associates to finite dimensional

vector spaces E and F the finite dimensional vector space

Hom(E, F) = L(E, F) (the space of linear maps from E to F), and associates to each pair of linear

maps f: E' - E and g: F -, F' the linear map Hom(E, F) -+ Hom(E', F') defined by h i-+ g It - f. Show that t is a smooth functor, contravariant in the first variable and covariant in the second. (Note: We can conclude from Exercise 7 that for each pair of vector bundles E, F over M,

Hom(E, F) = L(E, F) = U L(EX, FF) x

is a vector bundle over M. However, this vector bundle should be distinguished from HOM(E, F). HOM(E, F) is the space of sections of Hom(E, F); see the discussion preceding Lemma 5.59.) 9.

If E and F are vector spaces then E* ® F - L(E, F) by a canonical

isomorphism such that i. ® w e E* (& F corresponds to the linear map E -+ F given by v --+ <)., v> w. Show that this induces an isomorphism

L(E,F)-E*®F for any two vector bundles E and F over M. 10.

Formulate the definition of a smooth functor r of two variables which is

(i) covariant in both variables (ii) contravariant in both variables. Show that in each case t can be extended to the category of vector bundles with properties analogous to (a)-(d) of Proposition 5.57. 11.

Let L2(E) denote the set of bilinear maps E x E -+ R.

(a) If E is any vector bundle over M, show that L2(E) = U= L2(E,,) has a natural vector bundle structure with properties analogous to Proposition 5.57 (a)-(d). (b) Show that the canonical isomorphism L2(E) - E* ® E* of vector spaces induces an isomorphism

L2(E) - E* ®E* of vector bundles over M. 12.

Let E be a vector bundle over M. Let v ... , v, be a local basis for E

on U:

Manifolds and vector bundles

161

(a) At each point x e U define the dual basis i.'(x), ... , i.'(x) e Ex by Q.'(x), vi(x)> = Sr;

Show that ..' is smooth and hence a section of E*lv. Hint: Consider the isomorphism cp defined by (28). What is the image of A (x) under the transpose map '(qps)? (b) As in (b) of the preceding exercise, we identify LZ(E) with E* ® E*. Show

that every section g ofLZ(E) can be written locally in the form gs = E gi,{x)Ai(x) (9 Ax), where gij e C'(U) are unique.

xEU

5.9 Homotopy property for vector bundles Let E be a vector bundle over M. A map s: A - E is said to be a C'° section defined on the closed set A c M if for every x e A there is an open set U = Us containing x and a Cr section s1: U E such that sllu,A = SI vtA. The following lemma, a smooth version of the Tietze extension theorem, shows that s can be smoothly extended to a section ss M - E defined on all M.

Lemma 5.58 (Smooth Tietze Extension Theorem) Let E be a vector bundle over M. Suppose that s: A -+ E is a smooth section defined on the closed set A, that is, s is smooth in the sense defined above and p o s is the identity on A. Then s can be smoothly extended to a section of E.

Proof Consider the open cover { Us; x e A} of A which is given by the definition of smoothness of A. If we include also the open set M \A, we obtain an open cover of M. Let { 1;} be a locally finite refinement of this cover, and let {pi} be a partition of unity subordinate to { 1;} with supp pi compact in V1. Let s, denote a C' section 1; -. El,,, so that si and s coincide on A n Y, (where si = 0 if A n Y = 0). Then we get a C°° section s, of E by defining if x e Vi s,(x) _ Jp'(X)s,{X) 0

if xOVi

Then s = Y, .'s, is a C" section of E, and it is an extension of s, for if x e A then

s(x) _

p,(x)s,(x) _

p,(x)s(x) = s(x)-

Remark In particular, the lemma can be applied when E is a trivial bundle M x R'. Thus if A is a closed set in a manifold M and f: A -+ R' is smooth,

then f has an extension to a smooth function f : U - R' where U D A is open.

For any two vector bundles E and F over M, we let Hom(E, F) be the vector bundle for which the fibre over x e M is the vector space of linear

Boundary value problems for elliptic systems

162

maps from E,, to F,

Hom(E,F)=UL(E.,F,) X

(see §5.8, Exercise 8). If f : M - Hom(E, F) is a section, then for each x e M we have

f

a local representation of the form f: U -+ L(R', R°) where f is smooth (r and d are the fibre dimensions of E and F, respectively). The smoothness requirement on f is the same, therefore, whether it is regarded as a section M - L(E, F) or as a homomorphism E - F. Hence C'°(M, Hom(E, F)) = HOM(E, F), that is, the sections of Hom(E, F) are precisely the homomorphisms from E to F. Lemma 5.59 Let Y be a submanifold of M, and E and F two vector bundles over M. Then any isomorphism cp: El,. - Fly extends to an isomorphism El u - Fl v for some open set U containing Y.

Proof Note that cp is a section of Hom(E, F)jr. Applying Lemma 5.58 we can extend cp to a section g of Hom(E, F). Let U be the subset of M consisting of points x for which gx is an isomorphism. Since GL(R") is open in L(R"),

U is open and contains Y.

Throughout the rest of this section, I = (-a, 1 + a) denotes an open interval containing the unit interval [0, 1].

Definition 5.60 Two smooth maps fo: N M and f,: N -+ M are said to be (smoothly) homotopic if there exists a smooth map /: N x I - M such that I (x, 0) = fa(x) and I (x, 1) = f, (x) for all x e N. Ordinarily, a homotopy is defined as a continuous map/: N x [0, 1] - M. However, if two smooth maps /o and f, are continuously homotopic then they are also smoothly homotopic (see §10.6) so there is no loss of generality

in restricting attention to smooth homotopies. Example Let,?: M -+ M and A: M -+ R be smooth maps. Then (S, A): M M x R is homotopic to the map (y, 0). The homotopy is/(x, r) = (y(x), tf(x)),

XEM, te1. In Lemma 5.61 and Theorem 5.62 we assume that the manifold M is compact. This is sufficient for our purposes, but these two results in fact hold for any (paracompact) manifold. See [Hi] or [Sp 2] for the proof in general.

For a fixed t e I, M x {t} is a submanifold of M x I, so it makes sense to consider the restriction EI v x i,F, and we regard it as a vector bundle over M.

Manifolds and vector bundles

163

Lemma 5.61 Let M be a (compact) manifold. Then for any vector bundle E

over M x I

Elm.o - EIM.l where = indicates an isomorphism as vector bundles over M.

Proof Fix to e I and define z: M x I -+ M x I by t(x, t) = (x, to). Also let i : M x to - M x I denote the inclusion map. The restrictions t-'EIM,,,Q

and

EIMX,o are isomorphic

since they are isomorphic to i -'t -' E and I -' E, respectively, and since toy =i. By Lemma 5.59 this isomorphism can be extended to an isomorphism

z-'Elu ' Elu

(35)

for some open set U z) M x to. Since we are assuming that M is compact, then U = M x (to - e, to + e) for some & > 0, and restricting (35) to the submanifolds M x t we obtain Elm., = EIM X,o

when It - tol < e

(regarded as vector bundles over M). This implies that the set I0 = {t a 1; Elm. o} is both open and closed in I. Since I is connected we have Elm X ,

10=1. Theorem 5.62 Let N and M be (compact) manifolds and let f;: N -+ M, i = 0, 1, be smoothly homotopic maps. Then / i 'E 2t / 0'E for any vector bundle E over M.

Proof Let f : N x I -+ M be a homotopy connecting f, and j0. Since f, = f f where I,: N x t - N x I denotes the inclusion map, we have

Is'E^-'f-'Elv.,

t=0,1

and, by virtue of Lemma 5.61, the restrictions/ -'El,., and/ - 'EINX 0 are isomorphic vector bundles over N. A manifold M is said to be contractible if there exists a point x0 e M and a smooth homotopy f: M X I -- M such that I (x, 1) = x and I (x, 0) = x0 for all x e M. In other words, the identity map on M and the constant map are homotopic. Corollary 5.63 If M is contractible, then any vector bundle over M is trivial.

For example, any star-shaped open set in R" is contractible. Recall that U c R" is star-shaped if there is a point x0 e U such that, for any x e U, the line segment from x to x0 lies in U.

164

Boundary value problems for elliptic systems

5.10 Riemannian and Hermitian metrics Let E be a vector bundle over M. We say that a Riemannian metric has been

defined on E if we have an inner product <, )x = <, )E= on each fibre Ex which is smooth in the sense that for any sections s and t e C'(M, E) the function x H <s(x), t(x))x

belongs to C°°(M). Note that < , )x satisfies the smoothness property if and only if g;,{x) = <s,{x), s/x))x belongs to C'(U) for any local basis s1..... s, for E on U. (This is a consequence of Corollary 5.49.)

Example The product bundle M x f8' has the Riemannian metric
vkWk

k-1

v,weR'

for each x e M. It follows that any trivial bundle E has a Riemannian metric. Indeed, if fi is a trivializing map E -' M x R' then by transporting the metric on M x f8' we get a metric on E:
v, w e Ex ..w> w)x = I3xv. fl/3..w> where Nx: Ex -+ R' is the map induced on the fibres by P. It is clear that <, )x is an inner product since fix is a linear isomorphism, and, further, the smoothness property holds.

Proposition 5.64 Every vector bundle E has a Riemannian metric.

Proof Let {Uj, p,} be a partition of unity such that Eau, is trivial and pi has compact support in U , . Introduce any Riemannian metric (, > , into EI u, and then let x =

P,(x).

v, w e Ex

where the ith term is taken to be 0 if x 0 Uj. The sum is well-defined since it is a finite sum for each x. Thus < , )x is bilinear and x > 0 since p,(x) > 0 for all i. Also p,(x) > 0 for some i, hence 0 if v # 0. The smoothness property holds for < , )x because it holds for each term in the sum and every point has a neighbourhood in which the sum is finite. Let E be a vector bundle over M. In view of §5.8, Exercise 11,

L2(E) = U L2(Ex) xe M

is a vector bundle over M. A section g: M - L2(E) of this vector bundle has a local representation of the form g: U -, L2(f8') where g is smooth (see

§5.8, Exercise 12). Since L2(f8'), the set of bilinear maps on f8', can be identified by a linear isomorphism with the set of r x r matrices, we find

Manifolds and vector bundles

165

that the local representation of g has the form O(x) = [g, (x)]

where g;; e C`(U).

It follows that the set of Riemannian metrics on E is the set of sections gx = < , >x of L2(E) such that gx is symmetric and positive definite on Ex for all x e M. An inner product < , > in a vector space E induces an isomorphism between

E and its dual, E*. We call the map E -+ E* defined by v r. the "index-lowering" operator. Its inverse, E* - E, is called the "index-raising" operator. We denote the index lowering and raising operators by the symbols (subscript) and * (superscript), respectively. That is,

X,, =<X,-> and S=<S",-> The reason for this terminology comes from the Einstein summation convention where the components of vectors X e E with respect to a basis are written as superscripts X', while components of covectors e E* are written as subscripts ; . Let [g;;] denote the matrix of < , > with respect to a basis v...... v, of E, that is, g, = . The index-lowering operator maps the vector X e E with components X' to the covector X,, a E* with components Xi = Y gi;X',

(EX'v1)# =EX;i.' where i.'..... i.' is the dual basis corresponding to v1.... , v,. Now, let [g'J] = [gij]-' denote the inverse matrix. The index-raising operator maps the covector e E* with components i to the vector 1; * e E with com-

ponents S' = 0 4

(E Si)')" = E VVI The inner product on E induces an inner product on E* by <se, p> _ >; it is not hard to verify that it has matrix [g')] with respect to the <S *, dual basis that is, g'' = . Thus, it'

lI

llz =

is the length of with respect to the induced inner product on E*. Definition 5.65 A Riemannian manifold is a manifold M with a Riemannian metric g on the tangent bundle TM. By virtue of Proposition 5.64, Riemannian metrics exist on any manifold. A Riemannian metric on TM leads to the important subject of Riemannian geometry, with geodesics, curvature, and so on. However, we omit such topics since we will have no occasion to apply them; we need Riemannian metrics for various technical reasons and in order to study the Laplace-de Rham operator on M. In view of the index-lowering operator TM - T*M defined above we have a one-to-one correspondence between vector fields X e C°°(M, TM)

Boundary value problems for elliptic systems

166

and covector fields (or 1-forms) X. e C°°(M, T*M), which in local coordinates takes the form

X =YX1

a

I--.X,M =>X1dx',

ax1

where X; = Y g1JXj and g;,{x) = x. The inverse of this correspondence is of course given by the index-raising operator, i.e.

X'=

g'jX1.

Now let E be a complex vector bundle over M, i.e. the fibres are vector spaces over C. A C" inner product (or Hermitian metric) on E is a C-inner product

(, )E

(36)

in each fibre E,, such that the map x - (s(x), t(x))F belongs to C'(M, C) for all sections s, t e C''(M, E). Every complex vector bundle has a C x inner product. This is proved just as for the real case: we start with the usual inner product in Cr, r

= Y L";

V,WEC',

k=1

then make use of local trivializations EIu, of E to get a C' inner product on EIL,, and finally piece them together with a partition of unity. A Hermitian bundle is a complex vector bundle E with a given C' inner product. The reason that we will need Hermitian bundles is the following. Given a measure dµ on M (see §6.5), one can define an inner product on C0 '(M, E), the set of smooth sections of E with compact support, by (s, t)L2(M.E) =

(S(x). t(x))E, dp(x),

s, t e C0 '(M, E)

Then LI(M, E) is the completion of C0 '(M, E) with respect to this inner product.

The Gram-Schmidt orthonormalization procedure can be applied in Hermitian bundles. Let s ... , s, e C°`(U, E) be a local basis for E on the open set U. Since sl(x), . . . , s,(x) a E. are linearly independent at each x e U, we can orthonormalize them in the usual manner. we define u = s,/11s111

and if u,..... u,_1 have already been defined we let j-1

uj = sj -

(u1i s1)s1,

uj = uj/ Ilu;ll

(37)

i=1

Then for each j = 1, ... , r and each x c- U, the vectors u1(x), ... , u,{x) a ES

are orthonormal and span the same subspace as do s1(x),... , sj(x). In particular,

u1(x),... , u,(x)

is an orthonormal basis for E,

for all x E U. Moreover, it is immediate from (37) and the assumed C'O

Manifolds and vector bundles

167

property of (36) that the sections u,, ... , u, are smooth, i.e. belong to C" (U, E).

In this way we have essentially proved the following lemma.

Lemma 5.66 For any Hermitian bundle E over M and any open set U c M such that Elu is trivial, there exists a trivialization

fl:Elu -+ U xC' such that for all x e U we have (v, w)E= = (/iv, ffw)e, for all v, w e E.

Proof Let u,, . . . , u, e C'°(U, E) be the sections that we constructed above

by the Gram-Schmidt process. and consider the map U x C' -, El u defined by r

(x, c1,...,c,) + I ccu,4x). i=I

This map is a vector bundle homomorphism and it is bijective, hence by Lemma 5.47 it has a smooth inverse. Then we let fi be this inverse; that fl

has the required property is due to the fact that u,(x), ... , u,(x) is an orthonormal basis of Ex for each x E U.

1.

Exercises Let E be a real vector bundle over M with Riemannian metric gx

(a) The unit ball, B(Ex), in the fibre over x is the set of v c- E. such that lvlx < 1. Show that the unit ball bundle B(E) = U B(Ex) x

is an open subset of E. (b) The unit sphere, S(Ex), in the fibre over x is the set of v e Ex such that Ivlx = 1. Show that the unit sphere bundle S(E) = U S(Ex) x

is a submanifold of E of codimension 1.

Hint: If U is an open set in M such that El v is trivial, use the Submersion Theorem 5.25 to show that S(E)lu is a submanifold of El u Let E be a real vector bundle over M with Riemannian metric gx = < ,fix Show that the map v -' v/(1 + lvlx)'12 is a diffeomorphism of E onto its unit ball bundle B(E). (It is not hard to find an explicit formula for the inverse.) 2.

Let E be a vector bundle over M with a given Riemannian metric. Show that the map E -+ E* defined by the index-lowering operator Ex -+ Es on each fibre is a vector bundle isomorphism. 3.

168

Boundary value problems for elliptic systems

Let E be a vector bundle over M with a given Riemannian metric. Let F be a sub-bundle of E and set 4.

F1 = U (F,,)1, xeaf

where (Fx)1 is the orthogonal space of Fx in E. Show that F1 is a sub-bundle

of E, and E=F®F1. 5. Let F and V sub-bundles of E such that E = F ® V. Show that there exists a Riemannian metric on E such that V = F.

6. Let Y c M be a submanifold, and let the tangent bundle TM be given some Riemannian metric. Show that the normal bundle v(Y) (see §5.7,

Exercise 1) is isomorphic to the sub-bundle (TY)1 = U,EY (T,.Y)1 5.11 Manifolds with boundary

Let R", denote the upper half-space {x e ft" I x" >, 0) with the topology induced from R", that is, U c A+ is open if and only if U = U, n A+ for some open set U, in 18". We call Int U = U n {x a R' I x" > 01 the interior of U and

aU=Un(xeR"Ix"=0}

the boundary of U (this is not the topological boundary of U in 18", only the part which meets 318+ = 11"-` x 0). A locally Euclidean space with boundary is obtained by piecing together

sets which are homeomorphic to open sets in 18" or in R+. To carry this definition over to manifolds we need a definition of local smoothness in R", to be used for overlap maps of charts. Definition 5.67 Let U e A"+ be an open set. A map f : U - I8k is called smooth

if for each x e U there exists an open neighbourhood U, of x in R" and a smooth map f,: U, -+ 18k such that

filvr,v,=flu'v, For each x e U, we define the derivative off to be

Df(x) = Dfl(x) (Of course, if OU = 0 the present definition of smoothness is the same as the one used previously.)

We must show that the definition of D f is independent of the choice of f,. This is obvious if x e Int U, while if x e 8U it amounts to showing that if 0: U, -, Pk is a smooth map with U, open in R" such that ' l u, n R . =0, then D/i(x) = 0 for all x e U, n dR+. But this follows by taking a sequence of points x" a Int(U1 n R+) such that x" - x. A map f : U - V is a difeomorphism between open sets U and V in A+ if there is a smooth map g: V - U which is an inverse of f.

Manifolds and vector bundles

169

It is clear that the open sets in half-spaces together with the smooth maps between them form a category in the sense that the composite of smooth maps is associative when defined, and for any open set U in R8+ the identity map : U -+ U is smooth. The diffeomorphisms are the isomorphisms in this category.

Lemma 5.68 Let U be open in 9+ and cp: U - R8+ a smooth map. Assume that for some x0 a U, we have cp(x0) a ORB+. If we write cp = (cp', cp") where q': U -+ R8"-1 and cp": U --+ R8+ then the derivative of cp at x0 has the form 0(p

D

Dcp(x0) = 1

\

ngon*(x0)

J,

where DDB"(xo) % 0,

(38)

and D' denotes the derivative in the first n - I variables and D. = a/ax". Further, if x0 a Int U then also D"cp"(x0) = 0. (Hence D(p(x0) is not invertible.)

Proof Since cp maps into A+ we have cp"(x) > 0 for all x e U. By hypothesis cp"(x0) = 0, so the partial derivatives of cp" are D;cpn(xo) = lim (gon(xo + he,) - (pn(x0))/h,

j = I__ , n

k-0

= lim lpn(xo + lie,)/h. h-0

Now, if x0 e Int U we can let h 0+ and h 0" from which it follows that D,cp"(x0) >, 0 and D,cp"(x0) <, 0, whence D; cp"(x0) = 0 for all j = I,-, n. If

we have x0 e aU then this holds only for j = 1, ... , n - 1, and we have D"cp"(x0) >, 0 by letting h -- 0+ only.

Remark The second statement in the lemma says that if cp preserves upper half-spaces and maps an interior point to the boundary then the derivative must be zero in the normal direction.

The derivative operator D satisfies a funetorial property, namely, if cp: U - V and f: V - W are smooth then D(f o tp)(x) = D f ((p(x)) o Dcp(x),

xeU

(39)

For x e Int U this formula is just the chain rule, while if x e aU it follows by taking a limit x" - x where xn a Int U.

Proposition 5.69 Let U be open in A+, V open in A"+ and cp: U V a diffeomorphism. Then cp restricts to difeomorphisms Int U -+ Int V and aU -+ aV (between open sets in R" and between open sets in

R"- 1, respectively).

Proof Since cp is a diffeomorphism, Dcp(x) is invertible for all x e U (this follows from (39) with f = (p-1). If x c- Int U, then cp(x) e Int V by Lemma 5.68. Hence cp(Int U) a Int V. By considering the inverse cp-1: Int V - Int U it also follows that cp(Int U) m Int V. Hence cp(Int U) = Int V. But then cp(aU) = aV as well. Thus cp restricts to smooth maps Int U - int V and

Boundary value problems for elliptic systems

170

aU -+ V. Since the same is true of the inverse qp-', it follows that these restrictions are diffeomorphisms.

This proposition shows that the boundary and interior of an open set in 68+ are differentiable invariants, so we can define a manifold with boundary

just as in Definition 5.11 except that the chart domains are now homeomorphic to open sets in I8" or in R'. (and the smoothness of the overlap maps in Definition 5.11(u) is understood in the sense of Definition 5.67). The boundary and interior of M are defined to be

aM = U {K-'(a(K(U))); (U, K) is a chart on M} and

Int M = U {K-'(Int(K(U))); (U, K) is a chart on M}, respectively. In other words, x e aM if and only if K(x) E a(K(U)) for some (and hence all) chart (U, x) at x, and similarly for x e Int M (see Fig. 5).

From now on, we use the term "manifold" to refer to the extended definition. If M is a manifold in the original sense then OM = 0, and we say that M is a manifold without boundary. The next theorem follows at once from Proposition 5.69. Theorem 5.70 If M is a manifold of dimension n, then its interior Int M and its boundary aM are manifolds without boundary of dimensions n and n - 1, respectively. If tp: M -- N is a difeomorphism, N being another manifold, then cp restricts to di(jeomorphisms Int M - Int N and OM -+ aN. Most of the concepts introduced previously go over without modification to manifolds with boundary. But there are some differences. For example, products M x N do not exist in this category; the product of two manifolds with boundary would be a "manifold with comer" which we do not wish

to introduce here. This is the reason why in the definition of a smooth homotopy in §5.8 we required the interval I to be open. If M is a manifold with boundary aM, then M x I is also a manifold with boundary OM x 1, and the homotopy results hold true as before (with the same proofs). The smoothness of a map f: M - N is defined just as in Definition 5.13, keeping in mind that the smoothness of local representatives at boundary points now refers to Definition 5.67. A map c: I M, where I is an open, half-open or closed interval in R, is said to be a curve at m if it is smooth,

0 e I and c(0) = m. Two curves c, and c2 at m are tangent at m if (K o c,)'(0) = (K o c2)'(0) for some (hence all) chart (U, K) at m. As before, a tangent vector at m E M is an equivalence class [c]," of curves

c at m (two curves at m being equivalent if they are tangent at m). The tangent space T,"M = {[c],"; c is a curve at m}

172

Boundary value problems for elliptic systems

I = (-d, 6) then [c]m is tangent to aM, that is, [c],K e Tm(aM). Having defined the tangent bundle, all our definitions and constructions given in previous sections carry over directly to manifolds with boundary, with the same proofs, as the reader can easily verify. 5.12 Tubular neighbourhoods and collars

Let Y c M be a submanifold, and assume that Y and M are manifolds without boundary. A tubular neighbourhood of Y in M is a pair (E, f), where

E is a vector bundle over Y with projection p: E -' Y, and f is a diffeomorphism

f:E -+U of E onto an open set U in M containing Y, such that f is the identity on Y(where Yis identified with the zero section of E). If we denote the elements of E as pairs (y, v) where v e E,,, the latter condition on f can be written in the form

fly, 0)=y

(ye Y)

We call f the tubular map. A slightly more general concept is that of a partial tubular neighbourhood of Y. This means a triple (E, Z, f) where E is a vector bundle over Y, Z is an open neighbourhood of the zero section in E, and

f:Z-+U is a diffeomorphism of Z onto an open set U in M containing Y such that fly, 0) = y for all y e Y. See Fig. 6. Proposition 5.71 Let p: E -+ Y be a vector bundle over Y and let Z e E be an open neighbourhood of the zero section. Then there exists a dieomorphism

of E onto an open subset Z, of Z containing the zero section such that: (i) h preserves the fibres of E, that is, p o h = p. (ii) h is the identity on the zero section, that is, h(y, 0) = (y, 0) for all y> e Y.

Proof Choose a Riemannian metric <, )x on E. Given a positive function a e C'°(Y), let E(a) denote the union of the open balls in E,, centred at the origin of radius a(x),

E(6)= U {v;veE.,,IvI,,
We now construct a function a such that E(a) a Z. For each x e Y, there exists an open neighbourhood Vx and a number ax > 0 such that the vectors in EIv. which are of length
Manifolds and vector bundles

173

Note that a(y) > 0 for ally E Y. Now let v e Er such that IvI , < a(y). Note that a(y) < ax);) where aX(;, denotes the maximum among the numbers ax,;)

such that pi(y) # 0 (there are only finitely many such indices i). Hence IvL < ax(;) and, since y e V j), it follows that v e Z. Thus E(a) a Z. In view of the diffeomorphism of E onto its unit ball bundle B(E) given in §5.10, Exercise 2, we obtain a diffeomorphism of E onto E(a) given by v I-* a( p(V))'

v

+

IVI2)112'

This completes the proof of the proposition since we can take Z, = E(a).

Corollary 5.72 Any partial tubular neighbourhood (E, Z, f) of Y in M contains a tubular neighbourhood in the following sense: there is a tubular neighbourhood (E, g) of Y in M such that g(E) c f(Z). Proof Let h: E -+ Z, be a diffeomorphism having the properties as stated in the proposition above. Then let g = f c h. Thus it suffices to prove the existence of a partial tubular neighbourhood. We prove existence only for the case when Y has codimension I in M and there exists a vector field X = 1(M) such that X(y) 0 Tr Y for all y e Y. The basic idea of the proof is to use the integral curves of X starting at the points of the submanifold Y to define the tubular neighbourhood. The existence of X is equivalent to assuming that the normal bundle of TY in TMI1 is trivial (see §5.6, Exercise 3). It is true that any submanifold of M has a tubular neighbourhood, but we will have no occasion to use the more general result. See [La 2] or [Hi] for the proof in general.

Theorem 5.73 Let Y be a submanifold of M of codiniension I such that d Y = c'M = 0. Then there exists a tubular neighbourhood of Y in M.

Proof As mentioned above we prove the theorem only in the case when there exists a vector field X e 1(M) such that X(y) tk Ty Y for all y E Y. Consider the integral curves cm of X through the points m e M. The flow F of X, F: (m, t) I- cm(t),

is a smooth map of , into M, where

is an open set in M x GB that contains

M x 0 (see §5.5). Now consider the restriction FIg,,(yxn) of the flow, in other words, the flow determined by the integral curves of X that begin at the points of Y. We claim that for some open set Z in Y x P such that Z D Y x 0, the flow F is a djeomorphism of Z onto an open set in M. Since F(y, 0) = y for all y e Y this will prove the existence of a partial tubular neighbourhood of Y, namely

(Y x P, Z, Flz)

174

Boundary value problems for elliptic systems

First we show that FI y K n is a local diffeomorphism at each (y, 0) E Y x 0. Let (U, K) be a chart at y with the submanifold property, i.e. K(U n Y) = K(U) n (I8"-' x 0). The local representative of X is Tic o X o K-' and has the form x " (x, X(x)) where X: K(U) -' Il" is a vector function. The flow FF of X' is a smooth map from an open set in h(U) x R to ic(U) and it satisfies the equations

dt F(x, t) = X(F(x, t)),

F(x, 0) = x

(40)

F is just the local representative of F, i.e. F(x, t) = ic(F(K-'(x), t)). Thus, it suffices to show that the local representative FIK(UIY)KR is a local difleomorphism at each point ((X', 0), 0) a K(U n Y) x 0. The derivative of FIK(v,y)KR as a map from an open set in 08"-' x R to I8" is (41)

(Ds F, D, F)

and at the point ((x', 0), 0) these partial derivatives are equal to (see (40))

DxF=("O'

I,

D,

0)

As X is not tangent to Y, wehavelX"(x, 0) # 0, hence the derivative (41) is an invertible map at the point ((x', 0), 0). By virtue of the Inverse Function Theorem we obtain the following result (after transporting F back to the manifold Y by means of the coordinate map):

for each y e OR there is an open set Z c Y x P containing (y, 0) and an open set U in M containing y such that FIZ: Z - U is a diffeomorphism.

By shrinking U, if necessary, we can assume the additional property: if

xeUn Vthen (x,0)eZ.

It remains to show how these local inverses can be pieced together to form

a global inverse of F defined on some neighbourhood of Y in M. We can find a locally finite open covering of Y by open sets U, in M such that for each i there are smooth maps

fi:Z1-iU, and g,:U,-'Z, where Z; is an open set in Y x R and j = FIZ,, g1 and f are inverses of each other, and f and g, are the identity on the points of Yin their respective

domains. (Y is viewed both as a subset of M and as the zero section in Y x R.) The last condition on g, implies that if x e U; n Y then (x, 0) e Z1. Now shrink { U;} to obtain a cover { t; } of Y by open sets of M such that V,. c U1 (by Lemma 5.44), and let V = U I;. Let W = {x e V; if x e V, n V, then g1(x) = g,{x)}

Then W certainly contains Y because g1 is the identity on U, n Y. We claim

that W is a neighbourhood of Y in M, i.e. contains an open set in M containing Y.

Manifolds and vector bundles

175

Let y E Y. There exists an open neighbourhood W. of y in M which meets

only a finite number of Y, say V,...... v,. We can assume that y belongs to each one of these sets, for if y 0 Y,,, simply replace WY by its intersection with the complement C [',-,, and repeat for each such k. Next, replacing W, by

its intersection with U,...... U,,, we can assume that yEWY

Shrinking W, further, if necessary, we find that g,...... g,, must agree on Ws, since the respective inverses f,...... are restrictions of F. Indeed, we have

g,,(y) = (y, 0) a Zj, for all j = 1, ... , r and hence, if W is chosen small enough, the image

gj,(H) c Z;,

for all j = l,...,r

Since F(gj,(x)) = x for all x e W,, and the restriction of F to ZZ, n n Z,, is one-to-one it follows that g,,(x) = ... = g,,(x) when x e Wy. Since Wy meets Y when i = i ... , i but not for any other value of i, it follows that Wye W. Let W' be the union of the Wy's for each y E Y, and Z the union of the open sets g,{W' n l;). Note that W' z) Y and Z Y x 0. Since W' e W we can define a smooth map G equal to g, on W' n V,. Also F(G(x)) = x when x e W', and if u = g,(x) for some x c- W' n U, then G(F(u)) = G(x) = g,(x) = u. Hence G and FIz: Z - W' are inverses of each other.

Collar of a boundary The boundary aM of a manifold cannot have a tubular neighbourhood in M. However, it has a "half-tubular" neighbourhood called a collar. A collar of 8M is a diffeomorphism f : 8M x [0, oo) - V onto an open set V in M containing 8M such that fly, 0) = y for all y e 8M. In the proof of the following theorem we make the assumption that M is a submanifold of a manifold M without boundary. This is done so that we can easily apply Theorem 5.73. The existence of M is no restriction for us since later in the book we will assume that M =!D where f is an open set in 18" with smooth boundary; in this case we can take M = I". We leave it to the interested reader to work

through the details of the proof of Theorem 5.74 without the a priori assumption that M exists. Theorem 5.74 Let M be a manifold with boundary. Then aM has a collar.

Proof Assume that M is embedded as a submanifold of a manifold, M, without boundary. Choose a vector field X: M -+ TM such that X(y) is inward-pointing for all y e OM (constructed via a partition of unity). Then extend X to a vector field on M. This is possible since local extensions exist by definition, so Lemma 5.58 guarantees existence of a global extension.

176

Boundary value problems for elliptic systems

Since aM is a submanifold of M, and both are manifolds without boundary, we can use Theorem 5.73. It follows that there is a tubular neighbourhood (E, f) of aM in Al. As in the proof of Theorem 5.73, E is a trivial vector bundle aM x R. Thus, we have a diffeomorphism

F:aMxR -*U such that F(y, 0) = y for all y e (M, and U is an open set in M. It follows that F((M x [0, cc)) is an open set in M, and, since M is a submanifold of M, the restriction f = F Jim x [0. 21

is a diffeomorphism of OM x [0, co) onto F((M x [0, x)). The map f is a collar of (M. If we had chosen X to be outward-pointing at each point of OM the collar

would have been a diffeomorphism f: OM x (-x. 0] -+ U.

Theorem 5.75 Let M be a manifold with boundary (aM #0) embedded as a submanifold of a manifold Al without boundary (aM = 0). Then there exists an open set U in Al containing (M, and a di.Qeomorphism

f: OM x (-1, l) - U such that for all y E (M, fly, 0) = v f((M x (-1, 0]) is an open set in M containing (M, and

f(aMx(0,1))rM=0. Proof In the proof of Theorem 5.74, take the vector field X to be outwardpointing at each point of OM and then just compose F with a diffeomorphism

R - (-1, 1), for instance, x -. x/(1 + x2)'' 2. Later on when we use this theorem, the diffeomorphism f will be fixed once and for all. Thus, points in M near the boundary OM can be identified

with (y, t), y e (M, -1 < t < 0 and the points in Al \M near the boundary are identified with (y, t), y e (M, 0 < t < 1. Also OM is identified with OM x 0, of course. We will usually refer to aM x (-1. 1) itself as the tubular neighbourhood and OM x (- 1, 0] as the collar.

Fig. 7. The double of a manifold.

Manifolds and vector bundles

177

The boundaryless double of a manifold

Let M be a manifold with (3M s 0. Two copies of M can be glued along the boundary 8M to form a manifold without boundary, called the double of M (see Fig. 7). We will need the double of a manifold in Chapter 11. Let the two copies of M be denoted M x 0 and M x 1. The (disjoint) union (M x 0) u (M x 1) is given the obvious topology where a set is open if its intersection with M x 0 and with M x I is open in M x 0 and in M x 1, respectively. The double of M, denoted M, is the union of M x 0 and

M x I with the points (x, 0) and (x, 1) identified whenever x e 8M. To be precise, we have the equivalence relation on (M x 0) u (M x 1) defined by

(x,i)~(3',j)

if eithery=xe8M,or y=xeM and i= j

and if is the quotient space. We use the notation [(x, i)] for the points of M. If x e Int M we write just (x, i) (since there is a unique representative in the equivalence class).

Let n: (M x 0) u (M x 1) -' M be the canonical map (x, i) '- [(x, i)]. We

give M the quotient topology: a set V c M is open if n-'(V) is open in (M x 0) u (M x 1). Lemma 5.76 The double M is a locally Euclidean Hausdorff space without boundary.

Proof The graph, r, of the equivalence relation is equal to the union of three closed sets: the diagonal in (M x 0) u (M x 0), the diagonal in (M x 1) v (M x 1) and the set {((x, 0), (x, 1)); x e CM). Hence r is closed. It is evident that it is an open map, i.e. n-'(n(W)) is open in (M x 0) u (M x 1) for any open set Win (M x 0) ' (M x 1), whence Al is Hausdorff by §5.2, Exercise 7. For every point x e 6M there is an open set U a x and a diffeomorphism K: U - V where V is an open set in the half-space R. Let r: R" -+ R" be the reflection in the plane x" = 0, that is, 3(x', x") = (x', -x"), and then define

the mapic:U -Vby

A(x) - ( K(X) r

if x = [(x, 0)] K(x) if z = [(x, 1)]

where U = n((U x 0) u (U x 1)) is an open set in M and V = V u t(V) is an open set in R". It is not hard to verify that k is a homeomorphism. Hence M is locally Euclidean, and has no boundary (each point has a neighbourhood homeomorphic to an open set in R"). Remark If we define gyp: R" - R" by O (x', x") = (x', xn) if x" > 0 and cp(Y, x") =

(x', -x.) if x" < 0, then the restriction of 9 to each half-space At is of class CX, but (p itself is not C'. A chart (U, K) on M induces a coordinate map is on M as in the proof above, and in the same way the chart (U, (0 o n) on M induces a coordinate map W o is on M, but the overlap map (qp o ic) k -' _ W

is not of class C.

178

Boundary value problems for elliptic systems

To define a C'° structure on Al for an arbitrary manifold, therefore, takes a little more work due to the necessity of verifying the smoothness of overlap maps. In order to define a C' structure on M we use a collar of OM in M. The existence of a collar means that we have a diffeomorphism a: W -* aM x [0, 1)

such that a(x) = (x, 0) for all x e 3M, where W is an open set in M containing 3M. This induces a map d: W - aM x (-1, 1) defined by a(z)

a(x) if x = [(x, 0)] (z(a(x)) if x = [(x, 1)]

and W = a((W x 0) u (W x 1)) is open in Al. Theorem 5.77 Let M be a manifold with OM 0 0, and W an open set in M containing OM on which there is a diffeomorphism a: W - OM x [0, 1) such that a(xj= (x, 0) for all x e M. Then there is a unique C°° structure on the double M such that both inclusions M M are C°°, and such that the map a: W OM x (-1, 1) induced by a is a diffeomorphism. Proof We take as coordinate maps on Al the following:

(U x 0, K) and (U x 1, K), if (U, K) is a chart on M with U c Int M

(x-'(U' x (-1, 1)), (K' x id) o a), if (U', K') is a chart on aM

where id denotes the identity map (-1, 1) -' (- 1, 1). It is straightforward to verify that the overlap of two such maps is of class C. The domains of these maps cover M so we have (an atlas for) a C'° structure on M. Exercises 1.

Suppose that a,: W, - aM x (-1, 1), i = 1, 2 are two diffeomorphisms

satisfying the conditions of Theorem 5.77.

(a) Show that the CO' structures on k determined by a, and a2 are diffeomorphic.

(b) Under what conditions are the two C'° structures the same? 2.

Let M and N be manifolds and f : M -+ N and g: M -+ N continuous

maps such that f(x) = g(x) for all x e M. Then there is an induced map h: M - N given by h([(x, 0)]) = [(f(x), 0)] and h([(x, 1)]) = [(g(x), 1)].

Show that h is continuous. If f and g are of class C', what additional conditions are required in order that h also be of class C'? Let M be a manifold with OM * 0. Show that there exists a vector field X e f(M) that is inward-pointing when restricted to 3M. 3.

Let (X, d) be a compact metric space and A e X a closed subset. Let f: X - S be a local homeomorphism such that f1,, is one to one. (a) Let K be the set of points in X x X such that x y and f (x) = fly). Show that K is compact. 4.

Manifolds and vector bundles

179

(b) Show that there is a neighbourhood U of A such that f lu is one to one. (Hint: Define b: K - R by S(x, y) = d(x, A) + d(y, A); since K is compact, there is an c > 0 such that 6 s on K.) An open set it in R" is said to have C°° boundary if for each xo a dig there is a C" function cp defined on an open set W (in R") containing xo such that cp(xo) = 0, grad cp(xo) 0 0, and Q n W = (x e W; (p(x) > 0} 5.

Let Cl be an open set in .gt" with C' boundary. Show that i2 is a

submanifold of if, with boundary af. Let Cl be an open set in R" with C`° boundary. Note that if r: S2 - R" is a vector field on Sl then it has flow F(x, t) = x + t v(x). Suppose now that ail is compact, and use Exercise 4 to give a short proof of Theorem 5.73 in a tubular neighbourhood in R". this context, i.e. that ahas Cl 6.

6 Differential forms

6.1 Differential forms Before introducing differential forms we need the following result, which is

just a restatement of Proposition 5.57 for the functor N. Recall that N is a smooth functor by virtue of §5.8, Exercise 3. Proposition 6.1 The functor At extends to the category of vector bundles over M. For any vector bundle E over M let NE = l)xe /\k(Ex). If f: E - F is a vector bundle homomorphisms define the map Af: AkE -+ nkF such that its value on the fibre over x is given by the map Ak(fx): nk(Ex) -+ nk(Fx). Then

NE has a natural topology and C' structure making it into a vector bundle with the following properties:

(a) If f E - F is a vector bundle homomorphism then nk(f) is a vector bundle homomorphism nk(E) -- nk(F). (b) nk(f ® g) = At(f) ° nk(g) whenever f o g is defined and nk(idE) = idnkiEi.

(c) If E = M x E is a trivial vector bundle then Ak(E) = M x nk(E) is a trivial vector bundle.

(d) If /: Y - M

is a smooth map between manifolds then

At(f -E) _

t 'nk(E) If E is a vector bundle over M, there is a natural map E= -AtE defined on the fibres over x, which is easily seen to be smooth. Indeed, locally on

an open set U in M where Elv is trivial,

UxEik' - UxNE given by

it

is expressed as the map

map Eiki *AkE is C°` because it is multilinear.

If v ... , vk is a local basis for E on U, then the sections x H Vi1(x) A ... A Vi,(x),

ii < ... < ik,

form a local basis for NE on U; these sections are smooth because they are a composition of smooth maps. In view of Corollary 5.49 applied to the bundle

NE, any section s e C'°(U, NE) can therefore be written in the form s(x) = Y si,...ik(x) vik(x) A ... A Vik(x)

where si, ik e C'(U) and the sum is taken over all indices i, < 180

< ik.

Differential forms

181

Let M be a manifold and consider the tangent bundle TM. We know from the general considerations in §5.8 that the dual bundle T*M = UXEM (TTM)* is a vector bundle over M, which we call the cotangent bundle.

Let us write out explicitly what the trivializing maps look like for T*M. TTM has the Let (U, x) = (U, x,, ... , x") be a chart for M, then TU = trivializing map f: TU - U x I8" defined by v '-. (x, dn(v)). Now let i.'. ... . J." be the dual basis corresponding to the standard basis in R", i.e. i.': IO" - R is the projection on the ith coordinate. Since'(#,)(;.') = i.' oIs = dx1 we see that dx...... dx" is a local basis for T*M on U

and is the dual basis corresponding to a/cal,... , alax". Thus T*M has the trivializing map T*U -- U x I8" given by n

c1,.... bn)

E i=1

A C" section of T*M is called a covector field on M. For example, if f e C »(M) then the differential df can be regarded as a section M - T*M defined by df(x) = df, YM e (TTM)*. It is clear that df e C'"(M, T*M) because locally in the coordinate chart (U, x,, ... , x") we have (see the text following Proposition 5.29)

df=Of

ax,

ex"

dx"

and Of /axi e C'°(U ). Thus df is a covector field. Definition 6.2 We define the set of differential forms of degree r on M to be

r(M) = C r(M, n'(T*M)), the C°` sections of the vector bundle n'(T*M). In other words an dorm w e (2'(M) is a C' map

w: M - n'(T*M)

such that

w(x) a n'(T, M)

for each x e M.

Note that AO(T*M) is the trivial bundle M x Fl, so a 0-form is just a C'° function on M. Also, the 1-forms are the covector fields since A'(T*M) = T*M. We can also define differential forms on any open set by replacing M with U. In view of the comments following Proposition 6.1, if (U, x...... x") is a < ir, n dxi,, i, < chart for M then A'(T*M) has the local basis dx,, n on U. Thus a differential k-form on U can be written uniquely in the form

w(x) =I f ,...i,(x) dxi, n ... A dxi.

where f,...,, e C°°(U) and the sum is taken over indices i, < short we often write w = Y ft dx,.

(1)

< i,. For

182

Boundary value problems for elliptic systems

By identifying the differential form dx; with the corresponding form dx;

on the open set u(U) a R", we see that (1) is the local representation of w e C'(M. A'(T*M)) as defined in Exercise 4 of §5.8. The set (Y(M) of r-forms is a vector space over R, with addition and scalar multiplication defined pointwise, and in fact it is a C (M )-module. The product of differential forms is also defined pointwise: if a and q are forms of degree r and s, respectively, on the manifold M (or any open set) then the wedge product a n ry is defined by (a A r!)(x) = a(x) A n(x)

XEM

Note that a n rl is smooth, which is clear from the local representations of a and ry, whence a A q E SY''(M ). If f is a form of degree 0 (a real-valued function) and a is a form of degree r then f A a = .fa,

where f a is the r-form on M given by (f a)(x) = f (x) a(x). The operation n is bilinear over R and associative since this holds pointwise. In fact n is bilinear over C°°(M). Let f2(M) denote the direct sum of SY(M), r = 0, 1, ... , n together with its structure as a module over C'(M), and with multiplication n extended componentwise to Q(M). We call Q(M) the algebra of differential forms on M, and it is an associative algebra over C'(M). 6.2 The exterior derivative d

We begin with the definition of differential forms on open sets in R" and define the exterior derivative of such forms. The exterior derivative satisfies certain properties which characterize it uniquely, and make it possible to define this operator on manifolds. Then We identify TmR" = R" by the canonical isomorphism dx...... dx" is the dual basis in (R")* corresponding to the standard basis e1,. .. , e" in R". The canonical isomorphism TmR" = R" means that we can also identify /V(T.*R") = A'((R")*) which we do from now on. Recall from §5.1 that A((R")*) is the algebra over R generated by dx,, . . . , dx" with the relations

dx,Adx;=-dxj Adx;. As a vector space over R, A((R")*) has basis 1, dx,, dx, n dxj, dx; n dx1 A dx,,, ... , dx, n

i<j Every

A dx"

i<j
e A'((R")*) can be uniquely written as

_

dx,, n ... A dx,,

where the sum is taken over indices i, < . . . < i,. The real numbers

Differential forms

183

(n)-dimensional are the coordinates of 5 with respect to the basis r {d;, A A dx,,; it < . < i,} of n'((R")*). The C' differential forms of degree r on an open set U e R" are the elements of the C'(U)-module fY(U) = C"(U) ® n'((Q8")*)

(tensor product of vector spaces over R)

R

If co is such a form it can be written uniquely as

w= Y f ,...;, dxj, A ... A dx;, where the coefficients f;,...j, are C°° functions on U. For short we also write w = Y f, dx, where I is a multi-index (i ... , i,) with i, < < i,. Note that

S2°(U)=C"(U), and fr(U) = 0 if r > n. The wedge product of two differential forms is defined by

2A1 =>f,g., dx,ndxlefr"(U) if

a = Y ff dx, e O.'(U) and q =

(- l)"q n a and

fang=an fq,

y, dx, e SY(U ).

We have a A q =

feC"°(U)

(2)

Then Q(U) = Q°(U) ® i2'(U) ®. ® Q"(U) is an algebra with product A defined in the obvious way (see §5.1).

Remark In the definition of the wedge product n one has to take care

of the ordering: I and J are ordered, but Y is not. For instance, if r.,

at = Yajdx,a(2'(U) and q =YbjdxjeW(U) then a A q= E (ajbj - ajbj) dx; A dxj. j<j For 0-forms (functions) f e i2°(U) = C x(U) we define d( = Y O flax, dxj

(3)

so that df is a one-form on U, dfei2'(U). (This agrees with the definition in §5.3.) The operator d can be extended to forms of degree > I as we show in the next proposition. Proposition 6.3 Let U be open in I8". The operator d extends uniquely to an operator d: S2'(U) . (Y+'(U) for each r = 0, 1, ... , n, satisfying the following properties: (1) d is an anti-derivation with respect to A, that is, d is R-linear and

d(anq)=dxnq+(-1)'aAdq foraefY(U)andgefr(U). (ii) d2=dod=0.

Boundary value problems for elliptic systems

184

Proof For uniqueness, note that if a = Y f, dx, e W U) then property (i) gives

dot =Zdf, n dx,+f,d(dx,) By properties (i) and (ii) we have d(dx,) = d(dxi, n

A dx,,) = 0, so in

view of (3) it follows that dx =

oL dxj n dxj

(4)

axj

To prove existence, if x = f, dx, is an r-form, r >, 1. we simply define doe by the formula (4) and show that d satisfies (i) and (ii). Let q = g, dx,, then

2A11=Yf,g,dx,Adx, I.,

and, by definition of d and properties of A, d(a A ry) _

_

d(f -g,) A dx, n dx, (g,'df, + f,-dg,) A dx, A dx,

dj,Adx,AYg,dx,+(-1)'Yf,dx,AYdg,Adx,

=dl nq+(-])'IAd>f Note that in the third equation we used (2) and the fact that dg, A dx, _ (-1)' dx, A dg,, since dx, = dxi, n A dxi, is an r-form. To verify (ii) we first show that if f e C'(U) then d(df) = 0. This follows since

d(df) = d

Of (Y-

ax,

dxj _ Z

f

oz

ax;axi

dx, n dxi = 0

by symmetry of the second partial derivatives and the fact that dxj A dx, _

-dx, n dx;. Now, if a = f, dx, is an r-form, then (ii) follows from property (i) since f, e C '(U) and xi...... x,, e C '(U) so that d(df,) = 0 and d(dx,,) = 0. The operator d is called exterior differentiation. The uniqueness of d in Proposition 6.3 implies that its definition is independent of the coordinate system. This makes it possible to define the exterior derivative operator on manifolds.

Let w e fY(M) where M is a manifold. If (U, +:) is a chart on M, then w has the local representation Y f, dx,, n A dx,,, where f, eCx(U), and we define dwI(u.,,) _

Of,

dx, A dx,, n ... A dx,,.

(5)

exi

the sum being taken over all indices i = 1, ... , it and it <

< i,. Now, if

(U'. K') is another chart, then, by virtue of the uniqueness stated in Proposition 6.3, we have dwl (u. u..) = dwl(vnu.K)

Differential forms

185

Definition 6.4 For each w e f1(M) we define the exterior derivative on M by dwl u = dwl (u.K

for each open set U that is the domain of a chart (U, x) on M.

As shown above, the exterior derivative dw is well-defined (indepen-

dent of the coordinates K). The local representations (5) show that dw: M -+ 8'+r(T*M) is a C' section, so that dw a SY+'(M). In this way we have defined an U8-linear operator d: SY(M)

r=0, 1,...,n.

fY+'(M) for each

Theorem 6.5 For each r = 0, 1, ... , n there exists an operator d:1Y(M) -' tY+'(M), satisfying the following properties:

(i) d is an anti-derivation with respect to n, that is, d is R-linear and

d(anq)=daAq+(-1)'aAdq for a e lY(M) and q e SY(M).

(ii) d2=did=0

By the manner in which d was defined, no additional proof is required here, since these properties hold in a collection of open sets that cover M. Since d: lY(M) - SY+'(M) is R -linear for r = 0, 1, ... , it induces a linear mapping, d: O (M) -' f](M), on the direct sums by

(o,e1Y(M),r=0, 1,...,n) Let tp: M N be a smooth map. We would like to define the pull-back

of a differential form a on N by the map q'. The tangent of tp is a vector bundle homomorphism, Tip: TM - TN, and hence we have a vector bundle homomorphism 'Tip: T*N - T*M given by the transpose map in each fibre, and then a homomorphism A'(`Ttp): A'(T*N) n'(T*M) (see Proposition 6.1). Definition 6.6 Let tp: M - N be a smooth map, and let a e SY(N). We define the pull-back of a by W as follows: tp*(a) = N(' TO) ° 0(. tP

(6)

(For the case r = 0, we have Q°(N) = C'(N) and tp*(a) = a o tp, coinciding with the definition in §5.5 for the pull-back of C'° functions)

Note that cp*(a) is a smooth map M -+ n'(T*M) (it is a composition of smooth maps) and tp*(x)r e n'(T, M) for each y e M. Hence the pull-back, rp*(a), is a differential form of degree r on M, i.e. q,*(a) e SY(M). Let us examine what tp*(a) looks like in local coordinates. Let (U, x1, . . . , be a chart f o r N and (V, y1, ... , a chart for M such that tp(V) c U. We claim that (f, ° tP) do,, n ... A dtp1, tP*(y fi dxr, ... A dxr,) where tpr = x, c V is the ith component of the function tpI v.

(7)

Boundary value problems for elliptic systems

186

First, let us show that '(T,q,)(dxj) = dcp,l,,. This follows directly from the

definition of the tangent map (Definition 5.20) and the differential of a function: '(T,(p)(dx,)[c], = dx, T,rp[c],

where [c], a T,M

= dx, [cp o c],4, = (xi o (P o c)' (0)

= dcp,[c],

Now,

to

(7), it suffices to consider a monomial term n dxi; For y e V, we have by definition (6) A ... A dx,,I(,)) *(a)s

verify

a = f, dx,, n

=f,((P(y))-'(T,-p)(dxt) A ... n'(7 p)(dxt,) and we get the expression (7) since'(T,gp)(dx,) = drp,lr

Theorem 6.7 Let rp: M - N be a Cm mapping of manifolds. Then the pull-back map p*: fr(N) -+ SY(M) has the following properties: (i) ((p o 0)* = r/i* o (p* whenever (p a 0 is defined

(ii) If a : M -+ M is the identity map then i*: f'(M) - i1(M) is also the identity.

Proof If we recall that 7((p a ai) = Ttp -To from Theorem 5.21, and the functorial property of A', Proposition 6.1, then (Y'ot )*(a) _

°0

= N('TO)°/V TO)°a°(P-0 (ii) is obvious.

Let gyp: M -+ N be a C' map. Since cp*: IY(N) -1Y(M) is R-linear for r = 0, 1, ... , it induces a linear mapping on the direct sums, (p*: Q(N) -+ QM).

Theorem 6.8 If cp: M - N is a C'° map then cp*: i2(N) -+ f2(M) is a homomorphism of differential algebras, that is, (i) !p*(a A ,) = !p*(a) A (p*(7) for a e 11(N) and n e Q3(N). (ii) cp*(da) = d((p*a) for a e IY(N).

Proof Property (i) follows from the corresponding pointwise result (see (13) in Chapter 5). Property (ii) follows from the result proved below which characterizes the pull-back map locally.

Proposition 6.9 Let U and V be open in R' and cp: V - U a smooth map. The pull-back map, cp*, on C °' functions (defined by rp*(f) = fo cp where

Differential forms

187

f e C '(U)) extends uniquely to an operator 4p*: IY(U) - cY(V), r = 0, 1, ... n satisfying the following properties: (i) (p*(a n rl) _ (P*a A 4p*ry for a e SY(U) and n e LT(U)

(ii) docp* = 4)*od Proof Clearly if tp* satisfies (i), (ii) then
(8)

where (p, = x, o tp is the ith component of the function V. This proves uniqueness. For existence, we define tp*: IY(V) -+ SY(U) by the formula (8).

The property (i) is evident. Since rp* is linear, it suffices to verify (ii) on monomial terms, a = f, dxi, A . A dx,,. Now dcp*a = d ((.fi ° 4P) dcp,, n ... A dq ,.)

=d(ft'(p) n d(p,, n... Ad4Pr. and

cp*da =

tp*(

_ =

Of,

axj

If,

ax

o

dx; A dx,, n

4p)

A

dxi,)

d4pt n d4,, A ... A dlp,,

d(fi°co)Ad(p n...ndcp,. Remark If r p : V

U is a diffeomorphism then y, = x, o c p (i = 1, 2, ... , n) are

new coordinates on U. It is property (ii) which shows that the definition of d is independent of the coordinate system. We conclude this section with the definition of the contraction of a vector field with a differential form. But first let us consider the case of a vector X e E and a p-form

a e ,P(E*), where E is a finite dimensional vector space over R. Due to the universal roperty of exterior products there exists a linear map ix: AP(E*) - AP-'(E*) such that its value on decomposable p-forms is x(al

...naP)_

1)"'a'(X)aI n...na

(9)

sincethe right-hand side is an alternating function of a', ... , aP. (The symbol

n above the element a` means that it is omitted.) We define ic = 0 if c e Ao(E*) = 68, and extend ix by linearity to the direct sum A(E*). Proposition 6.10 Let E be a finite dimensional real vector space and let X e E. There is a unique linear operator ix: A(E*) - A(E*) which sends AP(E*) to AP'(E*), p = 1, 2, ... , and satisfies the following properties:

(i) ixc=0ifceR (ii) ix A = /'.(X) if A. e E*

Boundary value problems for elliptic systems

188

(iii) ix is an anti-derivation with respect to n, that is, ix is linear and ix(a n q) = ixa A q + (- l)Pa n ixry for all a e AP(E*) and q e Proof The uniqueness of ix is evident, since every element of AP(E*) is a sum of decomposable elements. The contraction operator ix defined above satisfies (i) and is linear. Also, when p = I the definition (9) is just (ii). It suffices to prove (iii) when a = a' n A ryq, where A aP and ry = ry' n x' and ry' a E*, because such elements span AP(E*) and respectively. Now, let aP+' =11' for i = I..... q. Then P+q

ix(anry)_

r=

(-1)'+1a;(X)a1 n...na n...nap+q

If we write the sum in two parts, Y_ P_, + DP+P+,, and in the second sum change the index of summation to i = i - p, we get (iii) because

(_

1)`+1 = (- l)p.(-1)r+1

Remark Properties (i)-(iii) imply that ix - ix = 0. Also note that the map (a, X) r-+ ixx is R-bilinear.

For a vector field X e .1(M) and a differential form a e S2P(M), ixa is defined pointwise:

(ixa)(m) = ix,.,a(m)

meM

We claim that ixa is smooth when X and a are smooth. In view of (1), it suffices to show this when a = dx,, n A dx,P in local coordinates By definition we have (U, x,, ... . p

ir(dx;, n ... A dx,P) = I (- l)" 'X" dx,, n ... A dY A ... A dx;p, r-1

which belongs to fP(U) = C'(U, AP(T*U)) because X'r = dx,,(X) e C°°(U). We call ixa the contraction or interior product of the vector field X with the differential form x. Sometimes it is more convenient to use the notation

aJX, instead of ixa.

The interior product has the following properties since they hold pointwise by Proposition 6.10.

Theorem 6.11 Let X e a1(M), a vector field on M. The operator ix:12(M) -+ Q(M) sends SZP(M) to QP-1(M), p = 1, 2, ... , and satisfies the following properties:

(i) ixf =0if feQ°(M)=C°`(M) (ii) ixw = w(X) if co e Q'(M). By w(X) we mean the function in Czi(M) defined by w(X)(m) = co(m)(X(m)).

(iii) ix is an anti-derivation with respect to n, that is, ix is a-linear and ix(a n'i) = ixx A ry + (-1)"a n ixry

for all aef2 (M) and geQq(M).

Differential forms

189

Exercises

Let E be a vector space over R. We showed in Theorem 5.9 that there is an isomorphism A (E*) = LL(E, R) which maps each decomposable element a' A . n a " to the alternating k-form (v,..... vk) " det(a', vj>. Show that if or e A°(E*), and iX a n°-'(E*) is identified with its image in L;-'(E, R), then 1.

ix a(v...... v _1)=a(X,v1,...,vp-1) Hint: First verify this when a is a decomposable element, then extend by linearity. 2.

Let cp: M -. N be a C' mapping. If X e.T(M) and Ye i(N) are

rp-related, i.e. Tip o X = Yo 4p (see §5.5, Exercise 4), show that

ixotp* = t0*-iy. Hint: It suffices to verify this for differential forms of degree 0 or 1 (because we can work locally). 3. Suppose that rp: M -' N is a diffeomorphism. Then i*..o 0p* = (p* o iY where rp* Y e X(N) is the pull-back of Y e .AT(M) (see §5.5).

63 The Poincare lemma A differential form w is said to be closed if dw = 0 and exact if w = da for some a. Since d o d = 0, every exact form is closed. The converse is not true as is clear from the elementary example

-y dx+ x2 +Y x

x2 + y2

dy

onM=R2\0

2

In this section we show that if M is contractible then every closed form on M is exact. Recall that a manifold M is contractible to a point ma e M if there is a C" function H: M x I -. M such that H(m, 1) = m and

H(m, 0) = mo for all m e M, where I = (-s, 1 +s) is an open interval containing [0, 1]. For t e [0, 1], we define ,: M - M x I by i,(m) = (m, t)

and we claim that is w if a form on M x I with dw = 0, then is exact (10) ijw - loco Assuming that (10) has been proved we can easily prove the Poincare Lemma. Theorem 6.12 Let M be contractible. Then any closed form on M is exact.

Proof Let a e IY(M). Then H*a is a form on M x I and in view of (10) we have

:;H*a-io*H*a=d7 for some q a SY(M). But observe that i*lH* _ (H o /l)* is the identity on M

190

Boundary value problems for elliptic systems

and io*H* _ (H ° r'o)* = 0 since H ° io is a constant map, so we obtain

x-0=dh Before proving (10) we need to develop a few preliminaries concerning the manifold M x 1. There is a global coordinate function, t, on M x I (namely, the projection M x I - I on the second factor). If (U, x1.... , is a chart

on M and it is the projection M x I - M on the second factor then (U x 1, x 1 ° t, ... , X.

7r, t)

is a chart on M x I. We will denote x; ° it by x;, and more generally if f is any C ' function on M, we denote the pull-back f ° n by f The dual basis of a/ar, a/a9...... a/5 ,, is given by dt, dz1, ... , d .in. Let i2'(M x I) denote the set of r-forms w e 12'(M x 1) such that locally in each chart we have who ,t =Y

n ... A dr(,

where the f's are C1 functions on U x I and the sum is taken over all multi-indices I = (il, ... , i,) with it < < i,. Note that if the f's were independent of t then w would be the pull-back by it of a differential form on M. We need an intrinsic characterization of Q'(M x 1). First note that there is a canonical isomorphism Tm,,)(M X I) - TmM X T,1,

so that T.M can be identified with the subspace T.M x 0,. Explicitly, this identification is obtained as follows: The tangent to the map :,: M -+ M x I is a linear map TmM - TmM x TI with image T.M x 0, and it sends the vector a/ax, I m e TmM to the vector a/az; I (m,,) a Tm,,)(M x 1). Similarly, we can identify T,1 with the subspace Om x T,I by mapping a/at 1, -- a/at I(m.t) With this understanding, we have, for each point (m, t) e M x 1, a direct sum of subspaces

T(m.,)(M X I) - (T,M x 0,) ®(0m X T 1)

Upon taking the dual of this isomorphism we obtain

T(,,)(M x 1) - (TmM x 0,)' ®(0m x T 1)°,

(11)

a direct sum of annihilators of T.M and T,1 in T(*,,,,(M x 1). It is not hard to verify (if one recalls the definition of a dual basis) that (TmM x 0,)°

has basis dtl(m.:)

(0m X TI)°

has basis d9,J(m.r), .... d ,I(,e.t)

Thus we see that fr(M x 1) consists of the C' sections of the vector bundle U(m.,) n'((Om x T,I)°), that is,

S2'(M x 1) = C'(M x 1, U A' ((Om x T l)')). (m.t)

Differential forms

191

Lemma 6.13 Let n: M x I -+ M be the projection on the first factor. If w e fY(M x I) then we have

co=w, +dt All for unique w, a fY(M x I) and q e fY-'(M x I). Proof There are really two issues to verify here. First of all we have to show that for any (m, t) e M x I we have wl(m.t) = to, + dtl(m.,) A it

for unique w, e n'((0m x T,1)°) and , e n'-'((Om x T! )'). Secondly, we must show that these forms co, and q which are uniquely defined at each point do indeed define smooth forms on M x 1. But, the existence and uniqueness of co, and q at each point is clear from (11) and Exercise 3 in §5.1. Locally, the existence and uniqueness of w, and q is also evident,

because then co is a sum of terms of the form f d9,, h - - - A dz;, and f dt h d.i,, h A &Z,, and f dt h &a,, h A d , _ , and we can separate -

off the terms containing the factor dt. The local expressions show that co, and q are smooth.

Remark Using the properties of the interior product in Theorem 6.11, we see that the unique q in Lemma 6.13 is given by q =

Now we define the operator K: fY(M x 1) - SY-'(M) as follows. If (o a SY(M x 1) we have w = (o, + dt h q where q e SY `(M x 1); for fixed m e M. the map t '--' o llm.f) =

is a smooth function on I with values in the finite dimensional vector space A'''(TmM) so its integral is well defined. We define Kwlm =

il I(m.,) dt

(12)

0

It is evident that Kwlm e A'-'(T* M) for each m e M, and we must show that

Kw is smooth. But it is clear from the local representation of elements of SY-'(M x I) that for all m e U (a chart domain), Kwlm is a sum of terms of the form f(m, t) dt f . dx,, h

C

. A dxi,_,,

where f e Cx(U x 1). Hence Kw is smooth, so Kw a SY`(M).

E

Theorem 6.14 We have on CY(M x 1)

iiw-iQw=d(Kw)+K(dw) Consequently, if w e (Y(M x 1) and do) = 0 then i i w - i ow = d(Kw), so that (10) holds.

192

Boundary value problems for elliptic systems

Proof The definition (12) is an intrinsic definition, independent of coordinates; to prove the theorem we can just as well work in a coordinate system (U x 1,

.x 1, ... , z,,, t). The operator K is clearly linear, so we need only consider two cases:

(1) w= fdzf, A...Adi,,= fdz,.Then dt n dz, + other terms

dw =

at so that by the definition of K Of

K(dw)I m =

(m, t) dt I dx,Im

Jot

C

f(m,/0)7

= [f(m, 1) = iiwl m -

dx,lm

io'wlm

Since Kw = 0, the theorem is proved in this case.

(2) w=fdtAdi,, n--.Adi,,_,=fdtAdi,.Then dw =

of j= r axj

dxj n dt n dz,

so that K(dw)Im = -

Of

I

jot \ fo,

dxj

(m, t) dt

dxj n dx,

(the negative sign arises because dxj n dt = -dt n dxj) and f (m, t) dt" A dx,

d(K(o) = d 0

_ Y

a

I

jmt axj \ f.'

f(m,r)dtldxjAdx,

I

C learly K(dw) + d(K(o) = 0 because we can differentiate under the integral sign. Since r 1 w = 0 and i ow = 0, the theorem is proved. Exercise Let f, g: M -, N be C" mappings which are homotopic, and let a be a closed

form on N. Prove that g`a - f a is an exact form on M. 6.4 Orientation of a vector bundle

Let E be a finite dimensional vector space over R. Two ordered bases t 1 , ... , v and vi, ... , v;, for E determine an isomorphism rp: E - E by

Differential forms

193

(p(v,) = vi. The matrix [ajj] of 4p is given by the equations n

vj = Y aij vi i=1

We say that v1, ... , v" and v;, ... , v. have the same orientation if det [ai j] > 0

and the opposite orientation if det[ajj] < 0. The relation of being equally oriented is clearly an equivalence relation, dividing the collection of all ordered bases of E into just two equivalence classes. Either of these two equivalence classes is called an orientation for E. The class to which v1. ... . v" belongs will be denoted by p. We also say that the ordered basis v1, ... , v" "belongs to the orientation", or is "positively oriented" with respect to p. An oriented vector space is a pair (E, p) where E is a vector space and p is

an orientation on E. If p is one orientation on E, the other orientation is denoted by - p. If (E, p) and (F, v) are two oriented vector spaces, an isomorphism cp: E -+ F is called orientation-preserving if it maps a positively oriented basis of (E, p) to a positively oriented basis of (F, Y). Of course, if this condition holds for one positively oriented basis of E, then it holds for all. We give R" the standard orientation, that is, the orientation which contains the standard basis e1, ... , e".

Definition 6.15 Let E be a vector bundle over M. An orientation p of E is a collection of orientations px for each fibre E,, which satisfy the following compatibility condition:

for any local trivialization /3: EIv - U x R" where U is an open, connected

set a M, the maps Qx: Ex - R" are either orientation-preserving for all x e U or orientation-reversing for all x e U.

If Q and i/i are two local trivializations E I v -' U x R", then the overlap map 1// o Q': U x R" -, U x R" has the form

Q-'(x, ej) _

i=1

arj(x)-(x, e1)

where det[ajj(x)] 0 0 for all x e U. Since U is connected this means that either the determinant is positive for all x e U, or negative for all x e U. Thus we can put Def. 6.15 in a more convenient form.

Definition 6.15' An orientation p of E is a collection of orientations pX for each fibre E. which satisfy the following compatibility condition: there is a collection { U; } of open, connected sets which cover M, and local

trivializations jlj: Elv, -+ Uj x R" such that Pi,,: E, - R" is orientationpreserving for all x e Uj.

For example, we can choose the Ui's such that they are the domains of charts in an atlas on M. A vector bundle is said to be orientable if it has an orientation. Note that if E is an orientable vector bundle over M, and M is connected, then E has just

two orientations; if p = { pX } is one orientation, the other is -,u = { -,u,).

194

Boundary value problems for elliptic systems

Examples

(1) Every trivial bundle is orientable. Indeed, let fl: E --+ M x R" be a trivialization. For the trivial bundle M x R" we can put the standard orientation on each fiber {x} x R", then choose the orientation µx on Ex to be such that fx is orientation-preserving for all x e M. (2) The Mobius band . K (see §5.7, Example (4)) is not orientable. Indeed, for each x = e2:a a S' we have just two vectors of "unit length" in the fibre

.-!!x, namely, [t, 1] and [t, - 1]. If .& were orientable then it would be possible to define a smooth nonvanishing section s: S' At by letting s(x) be the unique vector of unit one-dimensional length such that s(x) a ./! is a positively oriented basis of the one-dimensional space ,C. But we have already seen that every smooth section of .1/ vanishes at some point. It is convenient to reformulate the definition of orientation using exterior products. Let n = dim E. A nonzero element, co, of the one-dimensional space n"(E*) is called a volume element. By §5.1 (14), to is identified with an alternating map E'"' - R and then we have w((P(e1), ... , (p(e")) = detq - w(e...... a")

(13)

for all W e L(E, E) and e, a E. It follows that two ordered bases v...... v" and vi, ... , v;, have the same orientation if and only if w(v,, ... , v") and w(v;, ... , v;,) have the same sign. The following proposition is a consequence of (13). Proposition 6.16 Let E be a vector space over R and let co be a volume element

on E. Then there is a unique orientation for E such that the basis v ... , v" is positively oriented i f and only i f w(v ... , v") > 0. Two volume elements to, and w2 are said to be equivalent if there is a number c > 0 such that w, = cwt. Note that the orientation for E defined by a volume element depends only on the equivalence class of the volume element. Thus, an equivalence class [co] of volume elements defines an orientation on E, the opposite orientation being [-w]. If v ... , v" is a positively oriented basis with respect to a given orientation A i.", where of E, then that orientation is the one induced by CO = J.' A

;...... i." is the dual basis corresponding to v...... v". Volume elements are determined only up to a positive constant. To specify

a volume element uniquely, we need additional structure such as an inner product on E.

Proposition 6.17 Let E be an oriented vector space over R and let <,> be an inner product on E. Then there exists a unique volume element p e n"(E*) such that p(e,,...,e") = 1

for all positively oriented orthonormal bases et, ... , e" of E. In fact, if A p". More generally, if p', . , µ" is the dual basis then p = µ' A v,, ... , v" is any positively oriented basis of E and %', ... , An is the dual basis

Differential forms

195

p = (det[])"2 A' A ... A AN

(14)

then

Proof Clearly the condition p(e,, . . . , e") = 1 uniquely determines p by multilinearity. If p'. ... . p" is the dual basis in E*, then e...... e" is positively A p" since by virtue of §5.1 (14) we oriented with respect to if = p' A have p(e,, . . . , e") = det[b,j] = 1 , which is > 0. Now let v,, ... , v" be any

other positively oriented basis of E (not necessarily orthonormal) and consider the map qp e GL(E) defined by cp(e,) = v,. We have vj =

a,je, for

some n x t matrix A = [a,j]. Then

a,ja,k' <e1, e,>

a,ja,k

and hence det[(vj, vk>] = det ATA = (detA)2

Therefore, (det[(vj, vk>])"2 = Idet Al, and since det A > 0 we have (det[])ij2 = det A. Now, since is a one-dimensional space, we have p = c A' A for some c e R, and then by §5.1 (14) we get

.

A A"

c = p(v,,... , v") = det[(p`, vj>] = det[a,j]. Thus, c = (det[])"2 which proves (14).

In summary, corresponding to each inner product <,> on E there is a unique n-form it such that p(v,,... , v") = (det[])`12

for each positively oriented basis v,,. .. , v" of E. We call p the canonical volume element determined by the given inner product and orientation on E.

The orientability of a vector bundle can now be expressed in a manner analogous to Proposition 6.26, by using the functor A . Let E be a vector bundle over M of rank n (of course, the rank need not be the same as the dimension of M). If there exists w e C'(M, such that w(x) # 0 everywhere, then for each x e M the nonzero w(x) e determines an orientation p: of E. by Proposition 6.16. It is easy to see that the collection of orientations {p=} satisfies the "compatibility condition" of Definition 6.15, thus { ,u,,) is an orientation of E. It follows that E is orientable whenever the vector bundle A"(E*) has a section which is nowhere 0. The converse also holds, as we show in the following proposition.

Proposition 6.18 Let E be a vector bundle over M of rank n. Then E is orientable if and only if the vector bundle A"(E*) has a section which is nowhere 0.

Proof Suppose there exists co e C '(M, A"(E ')) such that w(x) # 0 everywhere. Then for each x e M the nonzero w(x) e A"(Ez) determines an orientation pX of E,, by Proposition 6.16. We claim that these orientations

196

Boundary value problems for elliptic systems

{lex} satisfy the "compatibility condition" of Definition 6.15'. Let xo e M and choose an open, connected set U a xo on which there exists a trivialization

/3: E I v- U x R'. The sections vi(x) = (fx)-'(e1),

i = 1, ... , n

form a local basis of E on U, and without loss of generality the basis {v1(xo),... , v"(xo)} for Ex0 belongs to the orientation pxo, that is wxo(v 1(x0), ... , v"(xo)) > 0 when x = x0. By continuity we then have wx(v1(x), ... , v"(x)) > 0

for all x e Uo

where Uo c U is an open set containing x0. This means 1'x: Ex - R" is orientation-preserving for all x e UO, and we have thus verified Definition 6.15', i.e. (,ux) is an orientation of E. It remains to prove the "only if" part. Suppose E is orientable, and let U be an open, connected set in M for which there is a trivialization ft: EI v - U x U8" such that the maps Yx are orientation-preserving f o r all x e U. Let e1, ... , e" be the standard basis in R". Then Ps '(e1), . . . , flx '(e") is a positively oriented

basis in Ex, and we let 1.i..... ).x be the corresponding dual basis in E*. It is easy to see that ;Y e C9(U, E*) (see §5.8, Exercise 11) and, if we define wv e C°°(U, n"E) by wv(x) = i.,' A . A ix, x E U, then v1, ... , v" is a positively oriented basis of Ex if and only if wv(vt..... v") > 0. Therefore, we can cover M by a collection of open, connected sets U; on which there exists w1 a C'(U;, n"(E*)) such that, for all x e U1, v") > 0 a v1,. .. , v" is a positively oriented basis of Ex. Let pi be a partition of unity subordinate to { U; }, and set

to = DIM. Because the supports of p1 are locally finite, this sum is finite in some neighbourhood of each point in M, whence w e C'°(M, A"(E*)). (As usual p1w1 is defined to be 0 outside U1, hence it is smoothly defined on all M.) At each point x e M, if v1, ... , v" is a positively oriented basis of Ex, we have (p4w1)(x)(v1, ... , v") >, 0 for all i and p1(x) > 0 for at least one index i. Thus, w(x) # 0. Since the fibres of the vector bundle n"(E*) are one-dimensional we can also write the proposition in the following way: E is orientable if and only if n"(E*) is a trivial vector bundle. Exercises

Let E and F be vector bundles over M. Show that if two of E, F and E ® F are orientable, so is the third. 1.

2.

Let /: N - M be a smooth map, and let E be a vector bundle over M.

Show that if E is orientable, then the pull-back bundle / -'E is also orientable. Find an example showing that the converse is not true.

Differential forms

197

Show that if E is any vector bundle (orientable or not), then E ® E is orientable. Hint: For any vector space E there is a "natural" orientation on 3.

E®E. 65 Orientation of a manifold

A manifold M is said to be orientable if its tangent bundle TM is an orientable vector bundle. It follows from Proposition 6.18 that M is orientable if and only if there is an n-form u e W(M) such that u(x) # 0 for

all xeM. Any two nowhere vanishing n-forms u, p' a f "(M) on an orientable manifold M of dimension n differ by a nowhere vanishing function: u = f p .

If M is connected, then f is either everywhere positive or everywhere negative. We say that p and u' are equivalent if f is positive. Thus on a connected orientable manifold M the nowhere vanishing n-forms fall into

two equivalence classes. Either class defines an orientation on M by Proposition 6.16, and these are the only two orientations on M. Examples

(1) The unit sphere S" c R"+' is orientable. Let n: S" -* R"+' be the unit normal vector field on S", i.e. (x) = x. The orientation on the tangent spaces Tx(S") is defined as follows. A basis v2, ... , v" in Tx(S") is positively oriented if and only if ,,(x), v,,. . ., v" is positively oriented in OB"+' with the standard orientation. We leave it as an exercise to show that this defines an orientation on the tangent bundle T(S") (i.e. satisfies the compatibility condition). (2) The real projective space 18P" (see §5.2, Exercise 9) is orientable if n is

odd, but not orientable if n is even. Let n: S" -, RP" be the canonical projection x i-+ [x]. Since it is a local diffeomorphism, the tangent map Tn: Tx(S") -- T,,J P") is an isomorphism for each x e S". Regarding T(S") as a sub-bundle of the trivial bundle T(R"+') = 68"+' x R"+', we have the map (T_xn)-toTxn given by (x,v)i-+(-x,-v) (15) from Tx(S") to T_x(S"). In view of the orientation defined on S" in Example (1), we see that this map is orientation-preserving if n is odd, but orientationreversing if n is even (since n + 1 is even and odd, respectively). In fact, this holds for any orientation on S" since S" is connected and there are only two

orientations on S", the standard orientation and the opposite orientation. If n is odd, we can define an orientation on each tangent space T,,,I P" by requiring that Txn be orientation-preserving. This definition is independent of the choice of the representative x in the class [x], and it clearly satisfies the compatibility condition of Definition 6.15 since it is a local diffeomorphism.

If n is even, then RP" cannot be endowed with an orientation for if this were possible then we could define an orientation on S" by requiring that Txn be orientation-preserving for each x e S'. But then (T_xn)-' o Txn would be orientation-preserving which contradicts (15).

198

Boundary value problems for elliptic systems

From the point of view of integration in §6.6 we will need to recast the definition of orientability in a more convenient form.

A diffeomorphism gyp: V - W between open sets in 31" is said to be orientation-preserving if

det Drp(x) > 0 for all

x e V,

i.e., the derivative map Drp(x):R" -+ R" is orientation-preserving for all x e V.

Theorem 6.19 Let M be a manifold. The following are equivalent:

(i) The tangent bundle TM is orientable. (ii) There is an atlas {(U,, K j)) on M such that all the overlap maps K, - cj' are orientation-preserving diffeomorphisms, i.e. det D(Ki D h; ') > 0 everywhere on n j(Uj n Ui).

Proof (ii) -+ (i) Suppose M has an oriented atlas {(UU,,c,)}. We define an

orientation µx on TIM, X E U,, as follows: A basis v...... v" of TIM is

"positively oriented" if dK;Is(v,), ... , dreiIx(v") is a positively oriented basis in

11". Since the overlap maps are orientation-preserving, it follows that the definition of µx is well-defined for all x e M. The collection of orientations {µx} obviously satisfies the compatibility condition of Definition 6.15, so TM is orientable. (ii) Conversely, suppose TM is orientable. Choose an atlas {(U,, r;,)} of M where the domains U; are connected. Then the maps dki`x: TIM -+ 18" (i)

are either orientation-preserving or orientation-reversing for all x e U,. In the latter case we can interchange any two component functions of t , to make it orientation-preserving. Thus, we can assume that the dKi's are orientation-preserving everywhere for all i. It follows that the overlap maps, hi 0 is ', are orientation-preserving. We say that an atlas {(U,, tci)} satisfying (ii) of Theorem 6.19 is an oriented atlas. An orientation on M is a maximal oriented atlas e?. Any chart (U, K)

that belongs to G is said to be positively oriented with respect to the orientation. We conclude this section by showing how a Riemannian metric on an oriented manifold determines a unique volume element.

Let M be a Riemannian manifold (see Definition 5.65), and assume that M is oriented. Then the inner product gs = <, )x determines a canonical volume element µs on TIM, and we obtain a map x e M -' µx a N(T1 M). This defines a differential form µ e f "(M) provided we can show that µ is smooth, and to do so we can work locally. Let (U, x1, ... , x") be a chart on M, then in view of Proposition 6.17 we have

µ = p'n dx, n ... A dx" in

U

where p:= det[g,j] and gij(x) =
Differential forms

199

If X is a vector field on M, we define the divergence of X to be the function div X e C°`(M) defined by

div X -p=d(pJX) (div X is well-defined since px 36 0, and N (T,*, M) is a one-dimensional space

for each x e M.) The smoothness of div X is a consequence of the local formula in the next proposition. Proposition 6.20 In a positively oriented chart (U, xl, ... , X

X=

X'afax1.

Proof The Riemannian metric and volume form have the local representa-

tions [g;j] and p = p"2 dx1 A ... A dx,,. where p = det[g;j]. Due to (9) we have _J

d(pJX)a (p12X')dx1n...Adx ax,

=

P-1/2Y a (p112 X+) . p ax,

Since div X p = d(p J X), the proposition follows.

Let f e C'(M). The gradient of f is defined to be the vector field grad f = X e X(M) defined by

df(Y) = g(X, Y).

Ye.Y(M)

In other words, grad f is the image of the I-form df by the index-raising (grad f )' a/axi

operator (see §5.10). In local coordinates, we have grad f = where (grad f )' = g'j of/ax .

The Laplacian of f is defined to be the divergence of grad f, A f = div(grad f) e C'(M(), and we have the local expression AJ=

p-"2

40

(p112g,,af/axj)

in

U

(3x

Exercises 1.

LetM and N be manifolds, and nM:M x N-4Mand nN:M x N -N

the projections onto the first and second factors, respectively. Show that

(a) T(M x N) 1v 1(TM) ®nN'(TN). (b) M x N is orientable if and only if both M and N are orientable.

Boundary value problems for elliptic systems

200 2.

Show that the torus S' x S' is an orientable manifold.

Let the Mobius band . Y be defined as in §5.7, Example (4), and let # be the canonical projection. Show that . /! is not an orientable manifold. Hint: If p e .Q'(,#) define f e C`°(682) by n *,u = f (dx A dy). Show that f (x + 1, - y) = -f (x, y), and conclude that f must vanish at 3.

n: R2

some point. 4.

Show that the sphere S' is orientable by using Proposition 6.19 (ii) and

the charts given in §5.2. Example (1). Show that the orientation on S" defined in this way coincides with the orientation defined in Example (1) above. 5.

Let M be a Riemannian manifold; denote the Riemannian metric by

9x = <, > for each x e M. Let f e C'(M) and suppose k e R is a regular

value of f Show that the vector field grad f is perpendicular to the manifold

f -'(k), i.e. for all x e f -'(k)
u e Ts (f -'(k))

Hint: v is the tangent vector to a curve c such that f (c(t)) = k for all t. Suppose M is an orientable manifold, f e C '(M) and k e R is a regular value of f Show that f -'(k) is an orientable manifold (if it is nonempty). 6.

Hint: Endow M with a Riemannian metric and let p be the canonical volume

form on M. Then let X = grad f in Lemma 6.21 (see below) to show that there exists a nowhere vanishing (n - 1)-form on f -'(k). 7.

Show that the sphere S" is an orientable manifold by constructing an

explicit n-form which is nowhere 0. Hint: Consider the volume form p = dx, A . A dx"+, in R"+' and see the preceding exercise. Let M be an oriented manifold with oriented atlas {(Ui, H,)} and let E be an oriented vector bundle over M with orientation {(U,, fi)}. Show that E can be made into an oriented manifold with orientation given by the atlas 8.

{(Eu1, Xi)} where

Xi=(r:,xid)cfi:Ev,

e,

Uix68

K, x id

Ki(U,)x68'cR"xR'

This is called the local product orientation on E. Addendum

In this addendum we prove a lemma which is useful for providing an orientation for a submanifold of an oriented manifold. Let E be a subspace of a vector space V. The transpose of the inclusion map e : E -+ V is the restriction map'e : V* E* given by A f-. A I E. Applying the functor A' to the restriction map we get a map A'(Y): A'(V*) - A'(E*) which we also denote by the symbol IE. By virtue of (31) in Chapter 5, we have (A'

forallA'eV*.

201

Differential forms

Lemma 6.21 Let p be a volume element on a vector space V. Let E be a subspace of codimension 1, and X a vector in V which is not in E. Then

(p1X)IE=0ifand only ifXEE. Proof Let X be a vector in V which does not belong to E. Choose a basis v1, ... , v" in V such that v1 = X and v2, ... , v" E E, and let ) ', ... , I" be

the corresponding dual basis. Since 0 # p E n"(V*), and n"(V*) is a one-dimensional space we may assume that p = I. (9) of the interior product, we have pJX=(_1)1+1I2

A

.

A I". By definition

A... A I.",

because i.'(X) = I and I.'(X) = 0 for i = 2, ... , n. Since PI E, ... , I."I E is a basis of E*, it follows that (p J X) I E 0 0. Conversely, if X e E, a similar method shows that (p J X) I E = 0. (E) is a volume element It follows that if X t E then (p J X)IE a on E. If V is given an inner product, then, by working with orthonormal bases, we obtain the following result. Corollary 6.22 In addition to the hypotheses of the Lemma, let V be given some inner product <, >, and choose an orientation on V and on E. If p (resp. a) denotes the canonical volume element on V (resp. E) determined by <, (resp. the restriction of <,> to E), then

a = (p1,)IE for a unique unit normal vector n E E. 6.6 Integration on manifolds The support of a differential form a e Q1c(M) is defined just as for functions,

i.e. supp a is the closure of the set of points x e M such that a(x) :0. We denote by £) (M) the set of k-forms a with compact support in M. If 8M # 0 we of course allow the possibility that supp a meets M. Although we did not mention it earlier, everything we have done in §§6.1 to 6.5 holds for manifolds with boundary. For points on the boundary, we have to use coordinate neighbourhoods which are diffeomorphic to open

sets in the half-space R+, but apart from that all the proofs remain the same (see the discussion in §5.11). The same holds for the definitions and

proofs in this section (which is important for §6.7 where we prove Stokes' theorem on a manifold with boundary), but to focus on the essential ideas we continue to write out the proofs under the assumption that the boundary of M is empty. The Change of Variables Theorem in R" will enable us to define the

integral of any differential form a e SR(M) (n = dim M) when M is an orientable manifold.

202

Boundary value problems for elliptic systems

Theorem 6.23 (Change of Variables) Let U and V be open in 68" and let (p: V

U be a C 1 diffeomorphism. Then for any integrable function f e L1(U) we have f o tp IJI a L1(V) and

1,

f(x) dx = f, f

IJQ(Y)I dy

v

where J,(y) = det DO(y) = ax/ay, the determinant of the Jacobian matrix.

Let U be an open set in 68". If a e f2,"(U) then a = f dxi n some function f e C'(U) with compact support. We let

A dx" for

Jarojf(xi,...,xn)dx u

The dx in the integral on the right-hand side denotes Lebesgue measure on 68". The order of dxi, ... , dx" in the Lebesgue integral makes no difference but it does in the differential form; for example in 682 we have Jfdx2 A dx1 =

_ffdxi

A dx2.

Now let gyp: V -. U be an orientation-preserving diffeomorphism between open sets in R". Let x1, . . . , x" and y1, ... , y" be the standard coordinate functions on U and V, respectively. Then cp, = x; 0 4p are the component functions of cp and we have

,p*(f dx1 A ... n dx") = f(cp(y)) dcpi n ... A dcp" = f(tp(y)) det Dco(y) dyi n ... A dy the second equality holding because of the alternating property of the wedge product. For any a e fr(U) with compact support we have 4ps(a) a f"(V)

with compact support equal to cp-1(supp a), and so by the Change of Variables Theorem 6.23 we have

tpta= a Iv fV If (U, ii) is a chart on a manifold M, we let ,c, = (K-')* denote the push forward of a differential form defined on U c M to a differential form defined on K(U) C R". Note that if a e ir,(U) then K,a E i2 (K(U)). Corollary 6.24 Let (U, K) and (V, 41) be charts on a manifold M such that the overlap map 0 c,c-i: K(U n V) - i/i(U n V) is orientation-preserving (in R"). If a e i2"(M) has compact support in U n V then Ks(a1 U) = J

J

+Is(aIv)

Now assume that M is an orientable manifold, and that some orientation has been selected. Let (U, K) be a positively oriented chart. If a e i2"(M) has

Differential forms

203

compact support inside U we define JM

a'=

J K(U)

K,(al U)

Due to Corollary 6.24, the definition of f Ma does not depend on the choice

of coordinate map K. Given an n-form a e f (M) with arbitrary compact support we then define its integral by

Yfu JM

i

c

where { p,} is a partition of unity subordinate to an oriented atlas {(U,, K,)}.

Proposition 6.25 The definition of I m a is independent of the oriented atlas {(U,, K,)} and the partition of unity { pi}.

Proof Let { V } be another oriented atlas of M (with respect to the same orientation!) and {X,} a partition of unity subordinate to { V }. Then Pia u,

= z f" PiXia t.i

since EJXi = 1

,

=EJ v,Pixia

since p,Xia has support in U, n V

i.i

since E,p, = 1

Remark If M is connected then there are just two orientations on M and changing the orientation reverses the sign of the integral f ma(. Densities

We assumed above that the manifold is oriented so that we could use the calculus of differential forms. But even on nonoriented manifolds it is possible

to integrate quantities known as "densities". To see how a density should be defined, let us investigate the behaviour of a linear functional A on C'(M) with respect to a change of coordinates. If (U, K) is a chart then A defines a linear functional AK on CC(K(U)) by C;.K>f> = <2,f ° K>, f E CC(K(U))

(f o K is defined as 0 outside U). Now, if (U', K') is another chart we have =

=<).J-(#e-K-')>, fECC(!c'(t 'n U))

For the sake of the argument let us suppose that i.,, = s'K(x) dx for some

Boundary value problems for elliptic systems

204

sK a C(K(U )); then we obtain

f e QV'), V

V

where V = K(U n U') and V' = K'(U n U') are open sets in R" and cp = K'- K

is a diffeomorphism between them. Now, by substituting

x' = cp(x) in the first integral, we see by the change of variables theorem that sK(x) = ssK.((p(x)). Idet D(p(x)I, or, expressed another way, the system of functions sK = sK o K E C(U., II') (where UK = U is the domain of K) satisfies the transformation property SK = Idet D(K o K'') o K I

sK.

in

U n U'

(16)

Now, in view of Proposition 5.47 (the text preceding it), there exists a line bundle having 1 x I transition matrices in

UnU'.

We denote this bundle by I A I and call it the density line bundle. In classical

notation we can also write h..K = Iax%axl, where ax'/ax is the Jacobian determinant for the coordinate transformation x -+ x'. A system of functions sK e C(UK, W) satisfying (16), i.e. .SK

hKK

is equivalent to having a section s e C(M, I AI), which we call a density. If s e C'(M, IAI) we say that s is a C" density. The condition (16) can be written in an invariant form by pulling it back to the density bundle. Let (U, K) = (U, x ... , x") be a chart with trivialization flu: IAI0 -' U x R. The constant function I e C°°(U, I8) gives us the

density flu' I E C'°(U, IAI) and we denote it by the symbol dx, - dx'. Pulling back sK a C(U, I8) by flu' we obtain, therefore, the density flu's,, = sK dx, ... dx" e C(U, IAI). Now let (U', x'1, ... , x,,) be another chart with a trivialization flu.: IAIu. -+ U' x R. Then (16) can be written sK = hKK.sK, where hKK = Since dxi . dx;:= flu.' 1 we have

dxi ... dx;, = Pull flu dxl ... dxM.

It is not hard to see that the map flu-" flu: U n U' by multiplication by the scalar function in

is given

thus

UnU',

which means that (16) can be written equivalently as in

UnU'.

The integral of a density is defined as follows. Choose an atlas {(U;, K;)} for M and a subordinate partition of unity {fir,}. If s e CC(M,Iis Al) a density with compact support, we set s t= (01 ° Ki ') JMiJR.,

sK,

dx,

(17)

205

Differential forms

where dx is the Lebesgue measure on R". Just as for n-forms it is not hard to show that the integral does not depend on the choice of atlas or partition of unity.

The measure associated with a Riemannian metric Now let M be a Riemannian manifold. The Riemannian metric g defines a Cm density. For every chart (U, x ,..., x") we let

p = Pu = (det[g;j])';2 in U where [g; j] = [g(a/ax;, a/axj)] is the local representative of the metric on the chart. This system of functions does indeed satisfy the transformation property for if (U', x1, ... , x;,) is another chart then

Pv/2=PYZlaxl Ox

in UnU'

by virtue of the proof of Proposition 6.17 since a/ox; = j a;j a/axj where a;j = ax-/ax,. We let d,u e C''(M,Idenote Al) the corresponding C'° density, i.e.

in

U

(18)

If f e CA(M) has compact support then f d p e C,(M, I Al) and we call fm f du the integral off with respect to dµ. In view of (17) and (18) we have

J f.dN=y M

(19)

i

Let X be a locally compact Hausdorff space. A linear map i.: QX) - C is said to be a positive functional if elf > 0 whenever f is real and f > 0. By the Riesz Representation Theorem (see [Ru 1]) there is a unique positive measure nt defined on the Borel sets in X such that

if=ffdin, feQX), and such that m(K) < oo for every compact set K c X. Furthermore, if X is v-compact then n is a regular measure. In particular, a manifold M is a locally compact and o-compact Hausdorff space to which we can apply this theorem. Let dµ a C(M, Ibe Al) a density

on M. It is not hard to see that the map f r j f du is a positive linear

functional on CA(M). By the Riesz Representation Theorem there is a unique regular Borel measure, in,,, such that

Jf dp =

$f dnt

when

feC,(M)

The space L,,(M, in,), p e R, consists of all measurable functions f such that If i P is integrable with respect to in, For p > 1, the norm

IIfII=(JlfIPdm)"

P

Boundary value problems for elliptic systems

206

makes LP(M, mµ) into a Banach space, where functions that differ only on a set of measure zero are identified. For p = 2, the space L2(M, is a Hilbert space with inner product U, 9)L M) =

f

fy dm H

From now on we always write dp rather than dm,,. Exercise 1.

Let d p e C(M, I A I) be a density and f E CC(M ). Verify that if U is an

open set containing the support of f then JM f du = jv f dp. (U is not necessarily a chart domain.) 2.

Let du e C(M, IAI) be a density. Verify that the map f '- f., f du is a

positive linear functional on CA(M). 6.7 Stokes' theorem

Let M be an oriented manifold with boundary OM. Before we can state Stokes' theorem we need to verify that the boundary of an oriented manifold

M is orientable, and then to choose the appropriate orientation on t'M. Recall from Proposition 5.69 that if rp: U -, V is a diffeomorphism between open sets in RR, then 4) restricts to a diffcomorphism
Dq(x) =

(x)

D

0

0.9.(x)

Now, if the Jacobian determinant of 4p is everywhere positive, det Dtp(x) > 0 for all

x e U,

it follows that det D'(p'(x) > 0 for all x e dU,

i.e. (p' also has positive Jacobian determinant everywhere. We claim that if M is orientable then its boundary OM is also orientable. To see this let {(U1, ic,)) be an oriented atlas on M, and restrict it to get an atlas on 8M

U;=U11- M, +:!=xilc,,,,,M Since the overlap maps for the atlas {(U;, h;)} have positive Jacobian determinant everywhere, the overlap maps for the atlas {(U1, At)} on OM also have positive Jacobian determinant everywhere. Now, let the upper half-space R = 1x: x" 0) in 68" be given the A dx". The boundary orientation on standard orientation dx, A An,. = R"-' = {x: x" = 01j is. by definition, the equivalence class of the volume form

a=(-1)"dx,A...Adxn-1

(20)

The factor (- 1)" is needed to make Stokes' theorem come out sign-free.

Differential forms

207

Let M be an oriented manifold, and let {(U,, r,)} be an oriented atlas on M.

The boundary orientation on am is defined by the following requirement: if : is an orientation-preserving diffeomorphism of some open set U in

M into the upper half-space R+ then the orientation on aU = OM r U is the equivalence class of k*a, that is, if m e aU then the basis v1, ... , v"-, e

Tm,(9M) is positively oriented if the local representatives v'; = dKI,"(vt), j = 1, ... , n - 1, satisfy

a(u1,.... r" ,) _

1)" det[ 0

(21)

By the discussion above, this orientation does not depend on the choice of local coordinates. Moreover, it is also clear from (21) (by continuity) that the "compatibility condition" of Definition 6.15' is satisfied, so we have indeed defined an orientation on T(aM).

Let v, e be any outward-pointing vector at in e OM. It is not hard to verify the following: a basis v2, ... , v" of T,(aM) is positively oriented if and only i f v1, v2, ... , V. is a positively oriented basis in TmM.

Example Let e1, e2, ... , e" be the standard basis on f8". If M is the halfspace {x a R"; x1 5 0}, then e2, .... e" is positively oriented in the boundary

orientation on M. M

We also have the following continuation of Theorem 5.70.

Theorem 6.26 If M and N are oriented manifolds and p: M -+ N is an orientation-preserving diffeomorphism then the induced map am - ON is also an orientation-preserving diffeomorphism with respect to the boundary orientations on OM and ON.

Now we are ready to prove Stokes' theorem. Theorem 6.27 Let M be an oriented manifold and or e fl,-1(M) have compact support. Let i : am -+ M denote the inclusion map. Then dx = SM

i*x SM

or, for short,

f da = J M

x

(22)

em

If M has no boundary then 1, da = 0. Proof Case 1. M is diffeomorphic to an open set U in 13". By definition of the integral we may assume M = U. Write

Boundary value problems for elliptic systems

208

a=> where the notation dxj indicates that dx, is omitted. Then dot= r=1

(-1)'-'

Of,

ax,

dx, n ... A d x,

so that

Jdi=J(_1)i_1idXi...dXn. ,_,

u

axi

Now, JAS of/ax; dx, = 0 since f, has compact support; thus

i. da = 0. Case 2. M is diffeomorphic to an open set U in A", and aU 0 0. Again we may assume M = U. The same result holds as in Case I except for the last term where we get by the fundamental theorem of calculus a

fO"

0) dxI ...

. dxe dx1..

axe

fR.-

since fe has compact support; thus dot =

1..

- fR^-'

(--1)e-'fe(x,0)dx1... dxe-1

On the other hand, since i*(dxe) = d(xe, (') = 0 on OR", we have

i*a =

J ou

i*a = fefe.

fe i r.

fe(x', 0) dx, n ... n

which is equal to Ju da by virtue of (20).

Case 3. M is any manifold. If p e C"(M) has its support contained inside some chart domain U, then the support of p a is a compact subset of U. Then

L d(pa) = f, d(pa) = feu

pa =

..LM pa

where the first and last equalities hold because of the support of p a, and the middle equality holds because of Case I or 2. Now if we choose an

oriented atlas on M and a subordinate partition of unity, then Stokes' formula (22) follows at once since both sides are linear in a. Remark There is also a version of Stokes' theorem on nonoriented manifolds which involves the integration of densities. See [BT].

Differential forms

209

Let M be an oriented Riemannian manifold. The Riemannian metric and orientation on the tangent bundle TM determine a volume form .Um a fr(M). Also, the restriction of the Riemannian metric to T(OM), together with the

boundary orientation on OM, determine a volume form uaM E Q"-'(aM). Let mM and meM denote the measures associated with um and uam. In the next theorem we apply Stokes' theorem to a Riemannian manifold and thus obtain the Gauss (or Divergence) Theorem. Theorem 6.28 Let M be an oriented Riemannian manifold. Then for any vector field X e X(M) with compact support we have

(div X) dmM = J (X, n> dmaM J where n is the outward-pointing unit normal vector along OM.

Proof We have div X - u = d(p J X), hence Stokes' theorem implies

j*(pIX)

J

I.

am

where I is the inclusion map OM -' M. For the right-hand side observe that X = (X n)n + v, where v is tangent along OM. In view of Lemma 6.21 and its corollary, we see that

*(u-1X)=(X'")uaM+0 and Gauss' Theorem follows at once. Remark We see from the proof that if X is an outward-pointing vector then

the boundary orientation on OM is the equivalence class of the volume element

i*(p J X)Efr-1(T*(OM)). 6.8 Differential operators in vector bundles

Let M be a manifold without boundary. Let E and F be complex vector bundles on M, and let A: C'(M, E) -+ C°°(M, F) be a linear operator. We say that A is a local operator if Au(xo) = 0 whenever u vanishes in a neighbourhood of x0. This implies that if u = 0 in an open set U then Au = 0

in U, so that supp Au c supp u for all u e C'°(M, E) We let Co (M, E) denote the sections u e C'°(M, E) with compact support.

Lemma 6.29 If A: C'°(M, E) - C°°(M, F) is a local operator then for any open set U e M there is a unique local operator Au: C°°(U, Eu) -+ C'°(U, Fv) such that Au(f J U) = (Af )I u

for all feCI(M,E).

(23)

Boundary value problems for elliptic systems

210

Proof Any f e C0 '(U, Ev) with compact support in U can be extended to a section f e C °°(M, E) by setting it equal to 0 outside U. In this way the equation (23) defines an operator Co(U, Ev) - C°°(U, Fv) which for convenience we denote by the same symbol A. Now, if f e C°°(U, Ev) we define Av f(x) = A(Of)(x),

x e U,

where 0 e Co (U, R) is a bump function equal to 1 on a neighbourhood of x. Since A is local, the value of A(i/t f)(x) does not depend on the choice of v. It is not hard to show that Au is local and satisfies (23). Uniqueness is also clear. We summarize the equation (23) by saying that "Av is equal to A on U ".

The same kind of argument shows that Au is equal to A. on U n V. Thus it makes sense to denote the restrictions Au by the same symbol A. We always do this from now on.

So let A: C°`(M, E) - C'(M, F) be a local operator and let (U, ,c) _ (U, x,, ... , x.) be local coordinates on M, and e...... ep and e'1, . . . , e, local bases of E and F on U, respectively. If u = F.J_, ufee is a section of Elv, then Au = y,=, (Au),e, is a section of Flv, and we have v

(Au)i = Y Aju;, i = 1__ , q i=I

(24)

where A;;: C'(U) - C'°(U) are linear operators. Before stating the definition of a differential operator we need the following

notion. Let (U, ,c) be a chart at xo. The partial differentiation operator asp, where ax, = 010x,, is defined on functions f e C °°(U, C) by transporting the partial derivatives of functions in C°°(K(U), C) using the chart x, that is, a°f - (&(f o r:- 1)) o K (the definition is chart-dependent of course). We say that f e C°°(M, C) vanishes to kth order at xo if a= = as;

a=f(xo) = 0 when

lad < k (for any chart at xo),

that is, all derivatives of order <, k vanish at xo. By the chain rule this latter condition makes intrinsic sense, independent of the chart at xo. If f e C°°(M, C") is a vector-valued function then it is said to vanish to kth order at xo if each of its component functions vanishes to kth order at xo.

Finally, if E is a complex vector bundle over M and f e C '(M, E), we say that f vanishes to kth order at xo if it does so in terms of local coordinates at xo, i.e. for each neighbourhood U of xo such that E I v is trivial:

the local representative J: U - C" vanishes to kth order at xo. This condition makes intrinsic sense, independent of the local trivialization of E (and chart at xo). Recall that the local representative f: U - C" is the f,,) where f; e C'°(U, C) are defined by f = Y fje;. vector function Definition 6.30 A kth order linear differential operator from E to F is a linear map A: C'°(M, E) -+ C°°(M, F) such that for all xo e M:

Au(xo) = 0

whenever u e C °°(M, E) vanishes to k th order at xo.

Differential forms

211

We denote by Diffk(E, F) the set of kth order differential operators from E to F. Note that if A E Diffk(E, F) then also A e Diffk+,(E, F). Evidently, a differential operator is a local operator.

Proposition 631 Let A be a local operator C°°(M, E) - C`°(M, F). A necessary and sufficient condition for A to be a kth order differential operator, A e Diffk(E, F),

is the following: for each chart (U, x...... on M the operators AU in equation (24) are kth order differential operators in the usual sense in the coordinates x1, ... , x8,

-

A,,(x, a) = Y app>(Y) - ... axi'

n

ax,',"

Proof The sufficiency of this condition is obvious. To prove necessity, we use Taylor's formula. By virtue of (24) we may assume that A C- Diffk(U x C,

U x C), i.e. a scalar operator. Let xo e U and x(xo) = 0. If f e C°°(U, C) then by transporting Taylor's formula for f -K-' by the chart K we obtain

f(x)

a!

a'f(xo)x' + Rk+ 1(x),

xeU

(25)

where

Rk+1(x) =

x'

gk,.(x)

and gk.8 a C°°(U, C).

Note: By x° we mean the function in C`°(U, I8) defined by xi' . x," where x1 e C'°(U, R) are the component functions of the chart K. Since x(xo) = 0 we have x;(xo) = 0 f o r i = 1, ... , n. Then x°, I a I = k + 1, is a product of k + I functions vanishing at xo, so Rk+, vanishes to kth order at xo. Since A e Diffk(U x C, U x C) then by applying A to (25) we obtain that Af(xo) =

JaY

a! aaf(xo) A(x2)(xo) + 0.

This holds for any xo a U; thus A = :ask a, a' where a, = A(x')/a!. Next, we define the principal symbol of a differential operator. We let p: T*M - M denote the cotangent bundle projection and then p-'E and p'F are the pull-backs of E and F to T*M. Recall that the set of vector bundle homomorphisms from p-'E to p-'F is the set of C' sections of

Hom(p-'E, p-'F), that is,

HOM(p-'E, p-'F) = C°°(T*M, Hom(p-'E, p-'F)). Definition 6.32 Let A e Diffk(E, F). The principal symbol of order k is the vector bundle homomorphism nMA a Cm(T*M, Hom(p-'E, p-'F)) defined as

212

Boundary value problems for elliptic systems

follows: for any (x, 1;) e T*M and c e E,, we let

akA(x, ) c = A(

(P

k.

u)(x)

(26)

where rp e C '(M) and u e C'°(M, E) are chosen such that cp(x) = 0, dxcp = S

and u(x) = c. The map irk: Diffk(E, F) -s C°(T*M, Hom(p-'E, p -IF)) is called the principal symbol map of order k.

Of course we must show that the definition of nkA is independent be local coordinates on of the choice of (p and u. Let (U, x,, ... , M such that E and F have local bases on U. The elements c e E. and trkA(x, S)c e FX can then be regarded as column vectors in Cp and CQ, respectively, and we have (26')

nkA(x, )c = Y aa(x) Sa' c I=I =k

where a= = [ajj']q -p is a smooth matrix function on U, S' = Si' Sam" and the ;'s are the coordinates of = El;, dxi. Since (26') does not depend on

the choice of rp and u, then neither does (26). At the same time, we have

shown that the local definition (26') is independent of the choice of coordinates and local bases in E and F. Furthermore, (26') shows that izkA(x, S) is a linear map Ex -+ F, and, since the expression on the right-hand

side depends smoothly on x and , we have defined a vector bundle homomorphism nkA: p-'E - p-'F, i.e. a C'° section from T*M to Hom(p-'E, p-'F). Remark The representation (26') corresponds to the local representation

A = [Ai,] _

(27)

aa(x) da, IalCk

in Proposition 6.31. We say that s e C'(T*M, Hom(p-'E, p- IF)) is a kth order homogeneous symbol if

for all i.>0. The set of all such symbols is denoted Symblk(E, F) and is a linear subspace

of C x(T*M, Hom(p-'E, p''F)). If A e Diffk(E, F) then (26') implies that akA a Symblk(E, F). Not every element of Symblk(E, F), however, is the principal symbol of a differential operator.

Let DSymblk(E, F) denote the set of homomorphisms s C'°(T*M, Hom(p-'E, p-'F)) such that for all x e M, s1: TX M -+ L(E,,, F,) is a homogeneous polynomial map as defined in §5.1, that is, sx a P'k'(T. M, L(EX, F1)). It is clear that nk maps Diffk(E, F) into DSymblk(E, F).

f2.

E. f= is called exact if the image of f is equal to the kernel of f,+, for Recall that a sequence of linear maps E,

i=1,2,...,n-1.

E2

213

Differential forms

Theorem 6.33 The following sequence of complex vector spaces and linear snaps is exact :

0 -+ Diffk _,(E, F) -. Diffk(E, F) - DSymblk(E, F) - 0 Proof The sequence is obviously exact at the first arrow since the inclusion map Diffk_,(E, F) -* Diffk(E, F) is injective. For the second arrow, if A E Diffk_,(E, F) then ttkA = 0; in fact by (26) we have ttkA(x, S)c = 0 since (pktt vanishes to (k - 1)th order at x. Conversely, if A E Diffk(E, F) then locally A has the representation 4

At, =

P

Aijuje;

where

Aij =

a;ji a=.

i=1 j=1

0 when IxI = k, and hence If nkA = 0, then (26') implies that A E Diffk _ 1(E, F) so we have exactness at the second arrow. To complete the proof we must show that the principal symbol map Jtk is surjective. Let s e DSymblk(E, F). We cover M with charts (U,, t:i) such that Eu, and F. are trivial. In local coordinates on U; we have T*(Ui)

Ui x88",

E. = U; x CP and F,,, ^-. U; x C9 and the local representation of s is a kth order homogeneous polynomial of S E 88" with q x p matrix coefficients. It follows that there exists A; E Diffk(Eu,, Fo) such that nkAI = s on the fibres of p-'E over U,. Now let I = Y iyi be a partition of unity subordinate to the cover U; and set A = Y O1 A,. Every point x E M has a neighbourhood that intersects only finitely many of the supports of i/i,, so it follows that A a Diffk(E, /F) and nkA(x, e) _

//

i,(x) . lrkAi(x, S) _

.Ii, Y

/ ,(x) . s(L, S) = s(x, S) //

xx

Definition 6.34 A differential operator A e Diffk(E, F) is elliptic at x E M if

nkA(x, S) is an isomorphism E. -+ Fx for all (x, S) e T*M\0.

The operator A is said to be elliptic if it is elliptic at each x e M. From now on we write it = ztk if no confusion is possible.

Let the tangent bundle TM be given some Riemannian metric g = < . >. This metric induces an "index lowering" operator TTM -* T*X M, defined by v i-4
ST*M = U {5 E T* M; < , S)x = 1), XeM

which is a submanifold of T*M (see §5.10, Exercise 1). Due to the homogeneity of the principal symbol, the differential operator A is elliptic if ttA(x, c) is an isomorphism for all (x, ) e ST*M.

Let (U, x1, ... , x") be a chart on M. With respect to the local basis O1Ox,,... , a1Ox" of TM, the Riemannian metric on TM has matrix [gij] where gij E C' (U) are given by

214

Boundary value problems for elliptic systems g,;(x) =
and, with respect to the local basis dx,,... , dx of T*M, the induced Riemannian metric on T*M has matrix [g''] _ [g;;] -'. We have ST*MI v =

U { E(M);

g''(X)55; = 1, where

Y Si dx,lx

xev

Let E be a Hermitian bundle, i.e. a C-vector bundle with a C" inner product (,)Ex on the fibers (see §5.10). Let dp be the measure on M induced

by the Riemannian metric g (see §6.5). We define an inner product on

Co(M,E)by

(u, V)L:(M.E)

f (u(x), r(x))E, dµ(x), u, v E Cp (M, E). M

LA(M, E) is defined to be the completion of Co (M, E) with respect to this inner product.

If E and F are Hermitian bundles and A:C-(M, E) C'(M, F) is a differential operator, then the (formal) adjoint A* is the linear operator CO '(M, F)

L2(M, E) defined by (u, A*v)L,(M.E) = (Au, v)L,(M.F)

(28)

for all ueCo(M,E) and vECo(M,F). We claim that the adjoint A* is also a differential operator. Since A* is a local operator (see Exercise 7), it suffices to show this in a chart (U, K) = (U, x1 , ... ,

on M for which Ev and F v are trivial. By virtue of the

Gram-Schmidt orthonormalization procedure, E and F have smooth local bases that are orthonormal for all x e U: (e,(x), e;(X))E, = big

and

(f (X), f;(X))F, = b,;

(29)

Now. if A has the local representation (27), then from (28) applied to u e C0 '(U, Ei,) and v e C0 '(U, Ev) it follows that the local representation of A* is A ' * u = Y (- l )121 Ial"k

1

a=(p a; u),

u E C x(U, CQ),

(30)

p

where p = det[g;;] is the Riemannian volume density and a= is the Hermitian

adjoint (conjugate transpose) of the matrix a,. Thus, A*u a C '(M, F), and since A* is local it extends in a unique way to an operator A*: C°`(M, F) -+ CI(M, E). Hence A* is a differential operator of order <, k. By considering just the highest order terms in (27), (30) we find that locally the principal symbols of A and A* are related by nkA*(X,

as(x) 5' _ (- 1)k[nk A(x, C )]k, I'I=k

and we have proved the first part of the following proposition, since to verify it, it suffices to do so locally.

215

Differential forms

Proposition 635 Let E and F he Hermitian bundles over M. If A e Diff"(E, F) then its adjoint A* a Diff"(F. E) has principal symbol

n" A *(x, b) _ (-1)"[n A (x, )]"

(31)

where the superscript b indicates the Hermitian adjoint Fx -+ EI of a linear map E., -+ F. Also, A** = A and if B e Diff".(E', E) then (A B)* = B* - A*. The proof of the second part is left as an exercise. Exercises

Let ; U; } be an open cover of M and let E and F be complex vector bundles over M. Given local operators 1.

A,: C'(U;, EU,) -. C'°(U,, Fe,)

with

AU,=AV, on

U; n UU

show that there exists a unique local operator A: Cx(M, E) -+ Cr'(M, F)

such that A;(f lu) = (Af)lu, for all f e C'(E). If all A, 's are kth order differential operators show that the same is true of A.

Show that a linear map A:C'°(M, E) -+ C'°(M. F) is a first-order differential operator if and only if the following holds: A(fg)(xo) = 0 2.

whenever f e C'(M, I8) and g e C°`(M, E) satisfy f(xo) = 0 and g(x0) = 0. 3.

Let X be a vector field on M. Then we have the derivation (or

directional derivative)

Dx:CZ(M,l)-+C'(M,R) defined by Dx f = Xf = df ' X. Show that Dx a Diff,(M x Q8, M x R) and find the principal symbol n,(Dx).

Let M be an oriented Riemannian manifold. The Riemannian metric g on TM induces a Riemannian metric on the cotangent bundle T*M 4.

by the index lowering operator defined in §5.10. Show that the Laplacian

A: CZ(M) -+ C'(M) is a second-order differential operator on M with principal symbol 11 112 = g(S, S) 5.

Show that if A e Diff,(E, F) and B e Diff,(E', E) then A o B e Difl,+,(E', F).

Prove that the principal symbol is multiplicative, that is, n,+,(A o B) _ 6.

Let A: C'°(M, E) -+ C' (M, F) be a differential operator of order k. Fix

(x, ) e T*M. Show that for any N E C'°(M) and u e C'(M, E) such that dep., = S and u(x) = c we have

nA(x, )c = lim t-"e-'*A(e'°tr)(x),

c e E.,

f- z,

Let E and F be Hermitian vector bundles over M. Show that if A: C'(M, E) -+ C'(M, F) is a differential operator then its adjoint A* 7.

defined by (28) is a local operator. If A has the local representation [A,)], k p

216

Boundary value problems for elliptic systems

(see (24)) with respect to the local bases (29), show that A* has the local representation 8.

Verify the local formula (30) for the adjoint of a differential operator.

9.

Prove Proposition 6.35.

10. Recall that the operators div and grad were defined at the end of §6.5. (a) Show that div: C'°(M, TM) -+ C'(M) and grad: C'(M) -+ Cx(M, TM) are first-order differential operators and (div)* = - grad. (b) The operators div and grad are operators on sections of the real vector

bundles TM and M x R. If we complexify the bundles to obtain Hermitian vector bundles TM ® C and M x C (see §5.1) then we have induced operators

div:C" (M,TM®C) -+ C" (M.C) and

grad: C"(M, C) -' C s(M. TM ® Q. Show that the formula (div)* = -grad still holds. 11. If E is a Hermitian bundle over M then we say that A E Diff2k(E, E) is strongly elliptic of order 2k if, for all (x, S) e 7*M\1 0, we have

((- I)kn2kA(x, S)v, v)E > 0 for all v e Ex, v o 0.

(a) Let A e Diffk(E, F). Show that A*A a Diff2k(E. E) is strongly elliptic if and only if nkA(x, S): Ex Fs is injective for all (x, S) E 7*M\,0.

(b) If E and F have the same rank (fibre dimension) show that A is kth order elliptic if and only if A*A is 2kth order strongly elliptic.

If I and J are ideals in a commutative ring R then I J denotes the ideal generated by all products fg, f e I, g e J. Clearly, I J is the set of all finite sums Y f g, where f, e I and g; e J. Since I(JK) = (IJ)K for any three ideals 1, J and K in R, it makes sense to define I' = I J. . . 1, and then I' is the ideal in R generated by products of k elements of I.

Note that C'(M, C) is a commutative ring, with addition and multiplication of functions defined pointwise. 12. Fix xQ a M, and let I, denote the ideal in C'i(M, C) consisting of the functions f e C'(M. C) vanishing at x0, i.e. f(xo) = 0. Then, f vanishes to

kth order at x0 if and only if f e I", that is, f is a finite sum of products of k + I functions in /_,,.

(Hint: The sufficiency of this condition is clear. To prove necessity. use Taylor's formula in local coordinates near x0.)

For the following exercise, recall that if.// is a module over R and I is an ideal in R then I .// denotes the submodule generated by the products a f, a e 1, f e . //; it consists of all finite sums l a, f, where a, E 1 and f, e _//. 13.

If E is a complex vector bundle over M, then C"(M, E) is a module

over the ring CY'(M, C), with addition and multiplication defined pointwise.

Deferential forms

217

Fix a point x e M, and let Zx denote the submodule of C°°(M, E) consisting of sections vanishing to kth order at x. Show that

Z= = I;+1 .

C°°(M, E).

14. Let E be a complex vector bundle over M. Define the complex vector space of "k jets" at x by

Jk(E)x = C'O(M, E)/Z'.'

where Zx is defined in Exercise 13. The canonical linear map C'°(M, E) Jk(E)x is denoted by f '- jk(f )x = [f ]x Show that the family of complex vector spaces Jk(E) = U Jk(E)x X

has a unique vector bundle structure such that jk(f) E C°°(M, Jk(E)) for all f e C m(M, E). Hint: For any trivialization 0: Ej v - U x C", define the map Jk(E)iu

U X/I=i
[f] x H (x, (aaJ (x))jaj¢k), X E U,

where f is the local representative of f Now apply Proposition 5.47. 6.9 The Hodge star operator and the Laplace-de Rham operator Let M be a manifold with some Riemannian metric g on TM. The goal of this section is to define the Laplace-de Rham operator on differential forms. But first it is necessary to define the Hodge star operator, and in order to do so we need to verify that the inner product <, )x on TXM induces in a natural way an inner product on the pth exterior product A"(7, M). It is convenient to consider this problem on the level of abstract inner product spaces, so that we can use orthonormal bases rather than being confined to the coordinate vectors a/x;. Thus let E be a vector space over R and let <,) be an inner product in E. From §5.10 we have the index-lowering operator *: E -+ E* given by v i- . Due to the functoriality of n°, there is an induced map A"(#):

M(E) -. n"(E*) such that S=wt^...nW

(32)

which we also refer to as the index-lowering operator. Let us examine how

this map looks in terms of a basis for E. Let v...... v be any basis for E. If S e A"E then S=

Y

tf,...ivV1D

A ... A v;v

(33)

for unique real numbers S'3--.'i'. We can extend the definition of the so that they are defined for all i1..... i" and alternating

coordinates

Boundary value problems for elliptic systems

218

(anti-symmetric) in those indices. Then C=

l 'i Pi

AV,,

'pv,D A

(sum over all i1,..., is).

(33')

Note that distinct indices it, ... , i,, can be ordered in p! ways and the terms

in the sum are a product of two alternating terms, hence symmetric in ii, ... , ip. Lemma 6.36 As usual, let g,j = and let A', ... , A" be the dual basis in E* corresponding to the given basis in E. Then S * e np(E*) is given by

_

Y_ ji<...<jp

(34)

;,...;,i.J' A ... A A",

where

i)

sum over all i

(35)

Proof The index-lowering operator E - E* is given by Y X'v, i-i

X12',

where X, = Y g,,X'.

In particular, the basis vector v, e E is mapped to Y g, j1) a E*, so the action of (32) on the corresponding basis vectors in APE is given by Vil n .. A Vip I L ghj, ... giv/DA1' A ... A )1v,

where the sum is taken over all j,,.. . , j p. Due to linearity, S in (33') is mapped to

S=

9i, 1,

E

E

L

L

J, <... <jp "all.

9iplp

i,..JPAJ,

9i,i, ' ' gipjp5

A ... A )j,

t,...rp )1, A

... A j"

This completes the proof.

Remark Note that the S j,...j,'s as defined by (35) are alternating in the indices j ... , jp so that we can also write (34) in the form 1

#=

P

E 51,...1, 1.1 i A ... A ,lj,.

(34')

We now wish to show that an inner product on E induces a natural inner product on APE, the pth exterior product.

Theorem 6.37 Let <,> be an inner product in the n-dimensional space E. There exists a unique inner product <, >(p, in npE such that ipi = det[lf;=i

(36)

219

Differential forms

for all vi and wj a E. This inner product is also uniquely determined by the fact

that for any orthonormal basis e .... e in E, the basis {e,, A i, < . < ip } is orthonormal in APE.

. A eip;

Proof The uniqueness of (, ),p, is clear. The existence of a bilinear map <,>1) satisfying (36) follows from the universal property for exterior products. Indeed, det[
is alternating and multilinear as a function of both (v1,. .. , v,) and (w1, ... , wp), so it defines an alternating p-multilinear map E(P) -, LQ(E, IB),

and by the universal property there is a uniquely determined linear map AP(E) -> La(E, R). This map can be viewed as an alternating p-multilinear map E'1" - (APE)* and hence there is a uniquely determined linear map APE -. (APE)*, i.e. a bilinear map APE x APE --. 88 such that A ... A wp)1- * det[]f .1

(v1 A ... A vp,

Now let us prove the second statement in the theorem concerning the orthonormal basis, namely <e,, n ... A e,p, ej, n ... A ejp>,p, =

6r1j1... S+PjP

(37)

By definition we have <e,, A ... A e,., ej, A ... A ejp>,P, = det[6,.jb]a.b=1

< j,,, we see that if ip # jp Keeping in mind that i, < < ip and j, < then either jp > ip so there is a column of 0's, or ip > jp in which case the pth row is 0. In either case, det[6j. j 1 = 0 if ip # jp. If ip = jp then expanding by cofactors about the last column or row we get D P-1 det[bt.jbla.b=1 =det[b,.jja.b=1

so it is evident that (37) follows by induction on p. To show that <, ),p) is positive definite, let x e APE and express it in terms of the orthonormal basis, i.e.

a = Y a1,..-',e n ... A e,p where the sum is taken over all indices i, < IPI_-

_

.

. < ip. By bilinearity, we have

ar,...rPaj,...jP31, J. (a1,...IP)2

whence ,p) = 0 if and only if

,

b tPly

3 0, 0 for all indices i1 < .

proves that <,)(p) is an inner product on APE.

< ip. This

220

Boundary value problems for elliptic systems

Corollary 6.38 Let v ... , v. be any basis for E, and let Vi, A

Y_ 1:

Vi,

P!

n=

P'

Y

Vi' A ... A via

be elements of ARE, with coordinates '--'P and n" -'P extended so that they are d e f i n e d f o r all i1, ... , iP and alternating in those indices. Then the value of

the inner product in Theorem 6.37 is (38)

P Proof By virtue of bilinearity and the definition of < I

_Sit...it,

1

y

,

>,P) we have

it...JP det[g,.j,]°.b= )

11

P! P!

because gij = .

<

1) 1)

P P

expanding the determinant we get

L L Y V*SP

i,...ipnj,...J, sgn o '

where SD is the group of permutations of { 1, ... , p}. Hence p. p. i...... it, cc sp

p! P! ;, ...

it, e E Sp

It,

which is equal to the right-hand side of (38).

Remark From now on we will denote the induced inner product on APE by <, > rather than <, >(,). Now, consider the exterior product AP (E*). An inner product <,> on E induces an inner product on E* by means of the index-lowering operator v i -. , and we denote it by the same symbol <, >. By applying Theorem

6.37 to this inner product on E*, we get the induced inner product <,) = <, >(p) in AP(E*), p >, 1. One could easily write down the analogue of (38) for this inner product, but the important point to remember here is that for any orthonormal basis µ ... , P. in E*, the basis it A ... n uiP i; <... < iP, /I

Differential forms

221

is orthonormal in A"(E*). If p = 0 we define <, > to be the obvious inner product on A0(E*) = R. Now we define the Hodge star operator. Proposition 6.39 Let E be an oriented n-dimensional rector space with inner product <, > and let u E A"(E*) denote the volume element on E. There is a unique isomorphism *: /\P(E*) A"-P(E*) satisfying x A *q = p

for

2, q E AP(E*)

(39)

Proof First we prove uniqueness of *. Let e,..... e" be an oriented orthonormal basis of E and p1, ... , u" the corresponding dual basis of E*. Then AP(E*) has the orthonormal basis µ,t n ... A ujp, j 1 < ... < JP, (40) and similarly for / -P(E*). The volume element for the given orientation is

u=u' n...nu".

Let tl=u'l A...Au'pwhere a=uit A...A pip where j, < < jr. Since = 0 unless (i...... ip) = (J1..... j,,) it follows from (39) that

*q = c u'P" A

A u'"

for some constant c

where i, <

< ip and i ,+ 1 < < i" are complementary sets of indices, i.e. the map is a i--* i, is a permutation of { 1, ... , n}. But then q A *q = c sgn(i)p

where sgn(i) denotes the sign of the permutation i. Since = I it follows from (39) that c = sgn(i), so that * is unique.

Now, define the operator *: AP(E*) -a n"-'(E*) by *(u" A ... A u'p) = A ... A u'"

(41)

where is a - is is a permutation of { 1, . . . , n} such that i, <

< ip and < i". Then (39) holds because it holds on the basis elements (40), and the operator * is an isomorphism because it maps a basis of /V(E*) to a basis of A"-"(E*). i,,+ 1 <

Remark The operators is characterized by the fact that * 1 = p, *u = 1, and for any positively oriented orthonormal basis u', ... , u" of E*

*(lii A...Alip)

=uP+1 A...A/4",

0
The operator * is called the Hodge star operator. By virtue of the symmetry of < , > we have for all a, rye AP(E*)

a n *q =
(42)

Proposition 6.40 The Hodge star operator * satisfies the following properties: *sa = a e AP(E*)

,

at, q e AP(E*)

Boundary value problems for elliptic systems

222

Proof Let is a - i, by a permutation of (1, ... , n) such that it < - . < iP and ip+1 <...
...

n-p n-p+l

n

Thus, c = (-I )P'"-P) sgn(i) and the first equation follows by using (41) again. For the second equation we have <*a, *q>, = *(X A **q = (- l )P("-p) *a n q

By virtue of the anti-commutativity of the wedge product A, the right-hand

side of this equation is equal to q n *a, or .u using (42). Hence = <*a, *q>-

Example Let e, = (1, 0,0), e, = (0. 1, 0) and e3 = (0, 0, 1) be the standard basis of f83 and dx, dy and dz the dual basis. Then the Hodge star operator on f83 (with the standard inner product and orientation) is given by *dx = dy n dz, *dy = -dx A dz, *dz = dx A dy This follows from the remark above.

The next proposition shows that the transpose of the interior multiplication operator ix: AP(E*) - AP-1(E*) is exterior multiplication by X,P. Proposition 6.41 Let E be a vector space with inner product < , >. Denote the induced inner product on each exterior product AP(E*) by the same symbol < , >. If X e E then X # n q>

(43)

for all aeAP(E*)and 1EAP-1(E*)

Proof Let et, ... , e" be an orthonormal basis of E and let p'..... µ" denote the (orthonormal) dual basis in E*. By linearity it suffices to verify (43) when

X = e; and a and q belong to the corresponding orthonormal bases in AP(E*) and nP-'(E*), i.e.

a=/1" n... A u',

and

q=fill A... A U1p

< iP andj1 < .. <j,-,. By definition of the interior product

where it <

(see (9)), we have 0

.

X

if i

e;

Since X = et then X. = p' and (43) is now easy to verify.

223

Differential forms

Remark The inner product can be extended to the direct sum A(E*) _ (B, ,PK ,,, AP(E*) as follows:

L

P.

a P, E qP ) _

P-1

P=


xP a AP(E*), qP c- n / \P(E*),

i.e. the inner product is extended by linearity such that APE is orthogonal

to NE if p : q. Then we have =

for all or c- A(E*) and q c A(E*).

Given a linear operator A: E - E, where (E, < , >) is an inner product space, then there is a transpose map 'A: E -' E. If we take the complexification E ® C then we get a C-linear operator A: E ® C -+ E ® C defined by A(v, + iv2) = Av, + iAv2, and with respect to the induced inner product on E ® C (see §5.1) the adjoint

map 'A: E ® C -+ E ® C is just the analogous C-extension of 'A, that is, 'A(v, + iv2) ='Av, + i'Av2. It follows that the spaces on which ix acts can be complexified without affecting Proposition 6.41. We complexify the real vector spaceAP(E*) by taking AP(E*) ® C, i.e. the space of "complex-valued" p-forms q = q, + iq2 where q,, 12 e AP(E*) are "real-valued" p-forms. The inner product < , > on AP(E*) is extended by C-linearity to obtain a Hermitian inner product

<x, + ia2' 11 + iq2> = <2,, qi>+ + i() (44) on the complex-valued forms. The interior multiplication operator ix, X e E, extends by C-linearity to an operator ix: AP(E*) ® C - AP-'(E*) 0 C. and its adjoint is exterior multiplication by X. as before. We also extend the Hodge star operator *: AP(E*) ®C -AP(E*) ® C by conjugate-linearity, i.e.

*(71 +iq2)=*1t-i*q21 so that (39) holds with the inner product (44). The properties in Proposition 6.40 continue to hold. The codifferentiai and the Laplace-de Rharn operator Let M be an oriented manifold, without boundary. We consider the following vector bundles over M:

AP(T*M), p=0,...,n and A(T*M) = ©AP(T*M), P=o

with the corresponding spaces of sections C'°(M, AP(T*M)), p = 0, ... , n,

and

C'(M, A(T*M)),

224

Boundary value problems for elliptic systems

i.e. the smooth differential forms on M (p is the degree of the form). For brevity we will usually denote these spaces simply C '(M, nil) and C'x(M, A), and in fact we will be dealing with the complexified spaces C°°(M, nil ® C)

and C'(M, A ® C). The exterior derivative d: C'(M, Ap® C) - Cx(M,

C)

on complex-valued forms is defined by d(ry, + i112) = dill + 'd72Let g be a Riemannian metric on TM, with local representation [g,j] with

respect to some coordinate chart (U, x1, ... , By virtue of the indexlowering operator TxM -+ Ts M, there is an induced inner product on TxM with local representation [g'j] = [g,,]-', the inverse matrix. By Theorem 6.37 and Corollary 6.38 we obtain an induced Riemannian metric on AP(T*M) defined as follows: if 1 I i;j,...;D dxi, n ... n dx;D,

P

dxj, n...ndxjy, P-

are elements of np(TXM) then

Y_ g"j,..,nn;,-Jo 1

P

(45)

The inner product (45) can be complexified as in (44) to obtain an inner

product on Ap(T, M) ®C. With this inner product, we then define the Hodge star operator

*: C-(M, AP ®C) - CNM, p ®C) pointwise at each point x e M, that is, (*)(x) = "a(x). The operator * has the properties stated in Propositions 6.39 and 6.40; in particular x,heC'(M,A"®C)

a A *q = <x,0,u

where p is the volume form on M induced by the Riemannian metric.

The inner product on AP(T"M) ® C induces an inner product on the sections C'(M, np(T*M) ® C) in the usual way (see §6.8),

(a, ,)=J aA*P M

=

I

JM


Definition 6.42 The cod ferential b: C°°(M, nil+' ® C) - C'(M, M (D C) is defined by

bq=(-I)"p+'*d*q if heC'°(M,C), and Of =0if feC(M,A°®C)=C°°(M,C).

Differential forms

225

Since d2 = 0 and ** is a constant multiple of the identity it follows that

62=0. The Laplace-de Rham operator A: C°°(M, A ® C) - C'(M, n° ® C) is defined by

A=(d+b)2=db+bd, and is a second-order differential operator since d and 6 are first-order operators. (This is clear from the local representations.) Also note that A preserves the degree of differential forms.

Theorem 6.43 The operators d and 6 are adjoints of each other. Further, at each point (x, i;) e TX M, we have

the principal symbol of d is exterior multiplication c - G A C the principal symbol of 6 is interior multiplication -i4. the principal symbol of A is multiplication by the scalar - IIIIZ

In particular, it follows that the Laplace operator A is strongly elliptic.

Proof By the anti-derivation property of d, and the fact that *b = (-1)°+' d*, we have d(a n *tl) = da A *71 + (-1)'a n d *t

=da n *17 +(-1)°(-1)°*'a n *brl = da n *q - Of n *6t1

whence by Stokes' theorem (since OM is empty) we have '

dan*el - J an*bq=0,

J

M

m

so that (da, q) = (a, 6q). This proves that d and 6 are adjoints of each other.

Since d is a first-order differential operator, the principal symbol of d is the map nd(x, S): A (TXM) ® C - M''(TXM) ® C given by nd(x, ) c = dxcp n co(x) + cp(x)dxw

=SAC where cp a C'°(M) and w e Cx(M, A"(T*M) (9C) are chosen so that p(x) = 0, d., p = S and w(x) = c. Since b is the adjoint of d, then nb(x,

-[nd(x, c)]

(see (31))

which by Proposition 6.41 is equal to -id.. Lastly, we determine the principal symbol of A. Since A = d o 6 + b o d, then nA(x,1;) = ird(x, S) o nb(x, S) + au(x, g) o trd(x, i;),

Boundary value problems for elliptic systems

226

whence ne(x, S): AP(TXM) ® C - np(TTM) ® C is the negative of the map

C i- % A i{.c + i{.( A C),

and by Lemma 6.44 it follows that ae(x, S)c = - II

112 . c.

Remark Since the principal symbol of a is - III 112, then in local coordinates

this means that if

w=>w;,...ipdx;, A . . . A dx;p then

ew =

(ew);,...;p dxi, A ... A drip,

where 2

(&o,,...ip = - g j i wi0A

i. j

8x, cxj

+ lower-order terms

To complete the proof of Theorem 6.43, it remains to prove the following lemma.

Lemma 6.44 Let E be a vector space with inner product < , ). Denote the induced inner product on E* by the same symbol < , ). Let E E* and let * e E be its image by the index-raising operator. Then a)=IISIl2

aEA°(E*)®C

(46)

where II II2 =

Proof Let v1.... , v be a basis of E and let

denote the corresponding dual basis. If e E* we have i; = Y_ S;1.i for some ; a R. Then c * = givi

where Si = C' g' ,. It suffices to prove (46) when at is a basis element, a = A" A A Aip, i, < ... < i,, since both sides of (46) are linear in a (note -

-

that a basis for a real vector space E is still a basis over C for the complexified

space E 0 C). In view of (9) we then have p

A i{.a = S

)r+

L, r=I

A ... A% A ... A Aip

and

A a) = Y Y 4iA a). i

j

by using the anti-derivation property from Theorem 6.11, ivj(i.' A a) = irj(A')a - )i A and keeping in mind that ai(vj) = 6ij we Also,

obtain t{ (S A a)

zz

zz

SiSioe +

xx

1)'J.' A ;." A ... A t A ... A )iv

=11SII2a-SAi{.a, which completes the proof of the lemma.

Differential forms

227

Example 6.45 Let (U, x1, ... , x,,) be a chart on M. Let us determine the formula in local coordinates for the codifferential on (real-valued) 1-forms,

b: C'(M, T*M) - C'(M) and the divergence of vector fields (see §6.5)

div: C' (M, TM) - Cx(M). We claim that bw = -P-,12 Y a (p112g1jwj),

w = > wj dxj

(47)

ax1

and

div X = -aX#, Note that X,1, _

X = Z X' a/axj.

X' dxj is the image of X by the index-lowering operator

TM - T*M.

By definition, with p = 1, we have bcw = - *d *w

(49)

Since *w =

w j*dxj we first need to work out the value of the Hodge star operator on the 1-forms dxj. Now, *dxj is an (n - 1)-form, so

*dxj=Yctjdx1A

A rXi A

Adx

for some functions cij on U. By definition of the * operator, see (42), we have dxj A *dxj = - p.

Since
(-1)`g''-p''2, thus *dxj= Since *w = we obtain

A dx,,, it follows that ctj =

(_1)igrj.p112dx1n...ndx n...ndx,,.

wj*dxj then, by applying the exterior derivative operator d,

d*w = Y a (g11p112ujj) dx1 A ... A dx,,. ,.jaxi Now *p = 1, so *(dx, n A p- 112, and (47) follows at once from (49). The formula (48) is now evident if we refer to the local expression for div X in Proposition 6.20. Example 6.46 Using Example 6.45 we can determine the value of A when it operates on functions f e Cx(M), i.e. p = 0. Since of = 0 then

Af = dOf + bdf = 6df.

Boundary value problems for elliptic systems

228

Because df is a 1-form then by (48) we have

Af = -div(df ) * = -div(grad f ). This is usually called the Laplace-Beltrami operator, see §6.5.

Example 6.47 Now we consider the Laplace-de Rham operator on p-forms in f1;". Let

w=fdx1A...Adxo.

With the usual inner product and orientation on fg", we have

8w= P

axj

j=1

and then taking the exterior derivative

2f

dow= -''P

a 2dxl

j=I axj p

2

"

+Y Y (-1)l+p+1axkaxj of dx l

p

ndxk

J=1 k=p+1

(50)

Further,

dw=

OfdxkAdx, k=p+Iaxk

=

r

k=p+1

so that

i

dw=

A...Adxp

(- 1)p of dx1 A ... A dxp A dxk axk

(-1)k+1

k=p+1

afdxp+I

n...nan...Adx"

axk

and then a2{

n

d*dw=-k,Y laxllidxp+l A...Adx" a2f dxjAdxy+1 + y y (- 1)k+1 axjaxk jI k=p+1

whence (-1)"p+1*d*dw

6dw =

a2f

"

-kap+1 axle dx1 A . . . A dxp

+ J-

+

1ke

(-1)j+p 1

a

axj

f dx, A &k

Differential forms

229

By adding (50) and (51) it follows that n a2f

Aw

=i axk2

dx,

Exercise Let

FF'3

a

1=1

axj

be a vector field in R', and let F, = Jj., F' dx; denote its image by the index-lowering operator which maps a/ax; -+ dx,. Show that the divergence and curl of this vector field are given by

div F = *d*(F,),

(curl F), = *d(F*).

Using the properties of * and d, verify the following vector identities:

(a) curl grad F = 0 (b) div curl F = 0 (c) curl curl F = -V2F + grad div F.

Part III Pseudo-Differential Operators and Elliptic Boundary Value Problems

231

7 Pseudo-differential operators on Di"

7.1 Some remarks about generalizing integrals While working with pseudo-differential operators our main tool will be the

Fourier transformation a, ('

i(k) = J

u(x) dx,

(1)

with the inverse a- 1, u(x) = (27r)' J

e'iz.c)

dS.

The Fourier transformation is well-defined on the L. Schwartz space, So(R"), i.e. the set of infinitely differentiable functions u e C`°(R") such that for all multi-indices a and fi,

sup Ix'(Deu)(x)I < oo. YE R^

Taking the maximum of these quantities for lal + I#I < k gives us a seminorm on E which we denote by Iul,.,. These semi-norms f o r k = 0, 1, 2, .. . define a topology on ,9'(R"), making it into a Frechet space. It is well-known that the Fourier transformation defines a linear, topological isomorphism aC:.5°(R") - .9'(R"), which can be extended to the dual space, a: 9"(l ") -

.So'(R"). If we denote by <,) the duality between .5' and .9', we have «, u> = <4), j5u> which is the definition of l5 on .50'. We wish to give a meaning to the integral in (1), and also to other similar integrals, in the case that the continuous function u(x) satisfies for some C > 0 and in > 0 the condition Iu(x)I < CO + Ixlz)m.

(2)

We do this by introducing a simple class of oscillatory integrals. For k > in + n/2, let

f e-r(x.4) u(Y) dx:=(l- Ag)k 233

f e-ux.s)

u(Y) Z k

(I + Ixl )

dx,

(3)

Boundary value problems for elliptic systems

234

where the derivatives in S are taken in the distributional sense. This definition is motivated by the formula Ix12)-k(l e-uX.41= (I + - A )k a-'X.4)

Since k > m + n/2, the integral on the right-hand side of (3) is absolutely convergent. To show that the definition (3) is independent of the choice of k, we look

at another way to define the oscillatory integral, using the duality of .9' and .9'. Choose X e .9' such that X(0) = 1 and set X,(x) = X(ex), E > 0. If u satisfies

the condition (2) then u e .9'. Since X. u - u in 9' as z - 0. it follows that

(Xeu)-.ain :9',i.e.foroe.9'and k>m+n/2,wehave = lim J(xu) e-o

= lim ffe_1o'x(sx)u(x)ai.i() dx ds

,-off

e-i(X.4)

u(x) (1 + Ix1z)k

JJ

(1 -

4

dx d

In the last step we integrated by parts, then let c -+ 0 which is justified by the Lebesgue dominated convergence theorem since the integrand is absolutely convergent. This shows that 0 is equal to the oscillatory integral $ e-ux.,9 u(x) dx defined in (3). In particular, (3) does not depend on the choice of k.

The second method of definition suggests quite generally that one may operate on oscillatory integrals just as with standard integrals, differentiation

can be performed under the integral sign, orders of integration can be interchanged and so on (these operations will be verified when the need for them arises). Example If u satisfies the condition (2), let us calculate of an oscillatory integral. Formally, we have

(t-1Ru)(x) = (2a)-" $Je

u in the sense

' 41u(Y) dy d

Now, consider e11X-7Mu(y) as a function of y,(with x as a parameter) and

regard $$. dy d as an oscillatory integral. Choose X e C0 '(R") such that X(0) = 1. Then

$Je1_7u(y) dy dg:= lim

(2n)-"

.J=

f $x(ey)z(s)

Jim Jx(sY)u(Y)(e

e-o = Jim

e-0

= u(),

as expected.

"u(Y) dY d

-1X)((x - Y)/e) dy

X(E(x - ez))u(x - EZ)(}j- 1X)(Z) dz

Pseudo-differential operators on R"

235

The purpose of this section has not been to prove any specific results; rather it has been to illustrate a method for generalizing integrals which will be used in §7.2. The following proposition is typical of the kind of oscillatory

integral considered there. For further details on oscillatory integrals, see [Ho 1] and [Shu]. Let t1' denote the set of all a e C°°(R" x R") such that IDSDXa(x, %)I C,.O(1 + ICI)' for all multi-indices a, f, and introduce the semi-norms

Ialk.c. = max sup ID,Dxa(x, )I (1 + ICI) -" I3+PI

k (x.f)

k=0, 1,2,.... ('Let a e O'". Then the value of the oscillatory integral

Proposition

=I

I e`4x.:)a(x, 4) dx d4

JJ

lim I J eux.e)X(ex, el:)a(x, ) dx

=-oJ

is independent of the choice of X e C0 '(R1 x R") satisfying X(0, 0) = 1. Further,

if M, N are even natural numbers such that M > n and N > m + n, then the oscillatory integral is equal to

I = JJex <xyMM[Na(x, )] dx ds,

(4)

where the integrand is in L1(R" x R"). Further, III 5 C-IaIM+N.c^. where the constant C depends only on M and N and not on the function "a".

Proof Let <x>2 = I + Ix12 and 2 = 1 - A, where A is the Laplace operator. Using the identities (x>-MM eiix., = eux.c) =

where M, N are even natural numbers, we can integrate by parts since X(ex, es) e Co to obtain Ia = $Je iix.a) X(rx, eo)a(x, S) dx di;

=

JJe"'<x> - M <>

M[x(x, e)a(x, f)]) dx ds.

It is not hard to see that the derivatives of X(ex, g) are bounded by a constant

independent of e, and for any real number s, each derivative of

is

certainly bounded by a constant times itself, so it follows that we can estimate the integrand by

I<x>-M