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+ 2<X,X> = 2
has a maximum at
f
X2f < 0
at
Thus, we
p0.
for any non-zero vector X E T
h(X.X) 1 0
obtain
po,
.
Consider
(M).
po
the restriction of
M
curvature of T'
to
h
T'
po
xT'
po
.
By assumption, the sectional
k = 0,
Thus, by equation (2.7) of Gauss with
.
X AY
is non-positive for any 2-plane
in
we have
0
g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y))
X, Y
where
are orthonormal vectors in
T'
,
By linearity,
.
0
this inequality holds for all
X, Y
in
T' p0
Thus, the
.
theorem follows from the following Lemma of T. Otsuki [1). Let
Lemma 2.1.
map and
h : Rk x Rk _, Rr
be a symmetric bilinear
a positive-definite inner product in
g
Rr
.
if
g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y))
for all X
X, Y
in Rk
,
in
then
Rk
and if
Ck xCk Cr-valued,
for all non-zero
r 2 k.
Proof: We extend map of
h(X,X) ql 0
Cr.
h
to a symmetric complex bilinear
Consider the equation
h(Z,Z) = 0.
Since
this equation is equivalent to a system of
h
is
r
quadratic equations:
h1(Z,Z) = 0,...,hr(Z,Z) = 0
.
123
§ 2. Fundamental Equations and Fundamental Theorems If
then the above system of equations has a non-zero
r < k,
solution
By assumption,
Z.
Z = X +
where
Y,
is not in
Z
X, Y
in
Rr
and
0 = h(Z,Z) = h(X,X) -h(Y,Y)+ 2
we have
h(X,X) = h(Y,Y) I 0
and
Rr.
Y ¢ O.
Thus
Since
h(X,Y)
h(X,Y) = O.
,
This is a
contradiction.
From Theorem 2.1 we obtain immediately the following. Theorem 2.2.
Every compact n-dimensional Riemannian manifold
of non-positive sectional curvature cannot be isometrically immersed into Remark 2.1. Kuiper (1].
R2n-1
Lemma 2.1 was conjectured by Chern and
They showed that it implied Theorem 2.1.
was then proved by Otsuki. T.A. Springer.
The proof above is due to
(See, Kobayashi and Nomizu [2].)
The lemma
4. Submanifolds
124
Submanifolds with Flat Normal Connection
03.
M be an n-dimensional submanifold of an m-dimensional
Let
Riemannian manifold
If the normal connection
M.
D
is flat,
we have
RD(X,Y) = DX Y - DYDX - D(X,Y) - 0 for any vector fields
X, Y
Proposition 3.1.
Let
tangent to
M
M.
be an n-dimensional submanifold
of an (n+ p)-dimensional Riemannian manifold normal connection p
M.
Then the
is flat if and only if there exist locally
D
orthonormal parallel normal vector fields. Proof.
If there exist p,
fields
locally.
Since
RD(X,Y)gr = 0.
Hence RD a 0.
orthonormal parallel normal vector
r
Then we have RD
is tensorial, this implies
Thus the normal connection
D
is flat.
Conversely, if the normal connection
(3.2) for any
Dgl = ... = Dgf = 0.
D
is flat, we have
DXDYgr - DY XSr -D (X,Y) 9r = 0 , p
orthonormal normal vector fields
gl,...,g
P.
We
put
£ er()g5
DXgr =
(3.3)
r = 1,...,P
,
s=1
where
er
are local 1-forms on
(3.3) in matrix form.
M.
In fact, let
For simplicity, we express
§ 3. Submanifolds with Flat Normal Connection
125
t9 = tS1,...,Sp) ,
(3.4)
Then (3.3) can be written as
DS = ®S
(3.5)
The matrix In terms of
0,
9
completely determines the connection
D.
(3.2) is given by
dO=82
(3.6)
Moreover, since
?,1,....Sp
are orthonormal, we also have
(3.7)
We need the following lemma. Lemma 3.1.
Let
8 = (8r)
be a
1-forms defined in a neighborhood of
(p xp)-matrix of 0
in
(3.6) and (3.7), then there exist a unique
of functions in a neighborhood of
A = -A-1(dA);
(3.8)
where
I
If
.
8
satisfies
(p xp)-matrix A
such that
AO = I;
tA = A-'
.
is the identity matrix.
Proof.
solutions.
0
Rn
(Uniqueness).
Then
Assume that
8 = -A-1dA = -B 1dB
A
and
B
are two
and AO = BO = I.
Thus
4. Submanifolds
126
d(AB) _ (dA)B-1 -A(B-1(dB)B 1) -A®B 1+ AB 1BOB- 1 = 0
Thus AB1
is constant.
Hence, by AO = B0 = I,
we obtain
A = B. (Existence).
We pass to
(n+ p2)-dimensional space
2
Rn+p
with coordinates
introduce the
and
x1,...,xn,zr (r,s = 1,...,p)
p2 1-forms which are coefficients of the matrix
Z = (zr)
A = dZ + Z® , Then we have
dA = dZA®+Zd® _ (A - ZC) A ®+Z®2
= AA0 . Thus, by Frobenius' theorem, hence, there is a matrix
that A = Z
is completely integrable and
of functions with AD = I
gives an integral manifold of the system
From this we obtain if we put
A
A
C = to-1,
dA = -A0.
Now, because
0
such A = 0.
is skew-symmetric,
then t
dC = -C(dtA)C = Ct®tAC = C 0
Thus, by the uniqueness, we obtain orthogonal.
This proves the lemma.
C = A,
i.e.,
A
is
§ 3. Submanifolds with Fiat Normal Connection
Applying the lemma to the normal connection a matrix
A
defined locally on
D,
we have
dA = -A®.
such that
Let
Then
A = (as).
dar = -E atwt
(3.9)
Put
M
127
gr = E argt.
Then
gl,...,gP
are orthonormal and
Dgr = E (dar + arwt)gs
(3.10)
Substituting (3.9) into (3.10) we find that
.
Dg{ =
= Dgr = 0. (Q.E.D.)
If the ambient space
M
is of constant curvature, then
we have the following result of Cartan [1). Proposition 3.2.
Let
of a Riemannian manifold M
M
be an n-dimensional submanifold of constant curvature.
Then the
normal connection is flat if and only if all the second fundamental tensors
are simultaneously diagonalizable.
A r
This proposition follows immediately from equation (2.9) of Ricci.
4. Submanifolds
128
Totally Umbilical Submanifolds
§4.
Rn
Let
xl,...,xn.
be the Euclidean n-space with natural coordinates
Then the Euclidean metric on Rn
is given by
go = (dxl)2 + ...+ (dxn)2 It is well-known that
is a complete, simply-
(Rf ,g0)
connected Riemannian manifold of curvature zero. We put
Rn(k) = ((x1,...,xn+ 1) E Rn+I
(4.1)
I.JT
((x1)2+ --- + (xn)2+ (sgn k)(xn+l)2
-2xn+1
where
sgn(k) = 1
or
-1
= 0,
according as
xn+1 2 0) k 2 0
)
,
or
k < 0.
The Riemannian connection induced by
go = (dxl)2+ ...+ (dxn)2+sgn(k)(dxn+1)2 on
of
Rn+1 k.
is the ordinary Euclidean connection for each value
In each case the metric tensor induced on
complete and of constant curvature Rn(k)
k.
Rn(k)
is
Moreover, each
is simply-connected.
A Riemannian manifold of constant curvature is called elliptic, hyperbolic or flat according as the sectional curvature is positive, negative or zero.
These spaces are real-space-forms.
Two complete, simply-connected real-space-forms of the same constant sectional curvature are isometric.
129
§4. Totally Umbilical Submanifolds
The hyperspheres in
Rn(k)
are those hypersurfaces
given by quadratic equations of the form;
fxl-al)2+ + (xn-an)2+sgn(k)(xn+1 a = (al,...,an+1)
where In
Rn(0)
,
an+l)2 = constants
is an arbitrary fixed point in
these are just the usual hyperspheres.
IR
,
n+ 1
Among
these hyperspheres the great hyperspheres are those sections of hyperplanes which pass through the center (0,...,O,sgn(k)/'qk{) k = 0,
Rn(k)
of
we consider the point at infinite on the
as the center in
Rn+1.
Rn+l
Great hyperspheres in
Rn(k)
Rn(k).
n+
-axis
is just a hyperplane in
Rn (0).
are totally geodesic hypersurfaces
All other hyperspheres in
Rn(k)
are called
Rn(0)
Small hyperspheres of
small hyperspheres.
x
For
The intersection of a hyperplane
through the center in
of
in R'1 , k y( 0.
are called
ordinary hyperspheres or simply hyperspheres if there is no confusion. Proposition 4.1.
submanifold
M
in the real-space-form
totally geodesic in hypersphere of an
of
Rm(k)
(4.2)
Rm(k)
Rm(k)
is either
or contained in a small
(n+ 1)-dimensional totally geodesic submanifold
.
Proof.
Rm(k),
An n-dimensional totally umbilical
If
M
is a totally umbilical submanifold of
then the second fundamental form
h(X,Y) = g(X,Y)H
h
satisfies
4. Submanifolds
130
for
X, Y
tangent to
Substituting this into equation
M.
(2.8) of Codazzi, we find
g(Y,Z)DXH = g(X,Z)DI
By choosing
the mean curvature and that
H = at.
DXg
g
DXH = 0.
Let
If
a unit normal vector field such
Then we have
a = 0,
is totally geodesic.
(Xa)g+ aDXg = 0.
(4.2)
implies that
Assume that
a ¢ 0.
orthonormal normal vector fields
m- n
be
a = JHJ
Since
are orthogonal, we see that the mean curvature
constant.
on
we obtain
Y = Z 1 X,
h = 0.
t
a
Thus
and is
M
Then we may choose
!l,. . 'gym-n
locally
M such that
(4.3) From (4.1) we find
2
(4.4)
Am-n = 0
Dtl = 0
(4.5)
.
Using (4.4), (4.5) and Weingarten's formula, we get
(4.6)
VX(g2 n ... A
where
Rm(k)
v .
tm-n) = 0
,
is the Riemannian connection of the ambient space
§ 4. Totally Umbilical Submanifolds
Equation (4.6) shows that the normal
Rm.
Euclidean m-space
subspace spanned by
Rm
Hence,
spanned by the tangent space
and the mean curvature vector
H
Rm,
dimensional linear subspace of
is a fixed (n+ 1) -
say
Let
Rn+1.
Rm.
be the position vector of
x = (xl,...,xn)
Rm.
is parallel in
g2,...,gm-n
the linear subspaces of Tp(M)
is the
Rm(0)
In this case,
k = 0.
Case (i).
131
Then, by (4.1)
and (4.5), we find
Y(x+ al) = v x-a-lAl(Y)+Dy(a lgl) = Y-Y = 0 , tangent to
for
Y
say
c.
Rn+l
Thus
M.
This shows that a-1
with radius
is contained in a hypersphere of
and center
c.
(resp., case (iii) k = -1).
k = 1
Case (ii).
M
is a constant vector,
x + a 1gl
simplicity, we consider the position vector
For
relative to the
x
center
(0,...,0,1)
(resp., (0,...,0,-1))
of
Rm(1)
(resp.,
Rm(-1))
in
Rm(1)
(resp.,
Rm(-1)),
to
Rm(1)
(resp.,
Rm(-1))
that
VWr1 = W
Rm(-1)
in
r = x in
for any vector W
where
Moreover, we have
v'
Rm+1
.
For each point
p
is a unit normal vector Rm+l.
It is easy to verify
tangent to
Rm(1)
is the Riemannian connection on
(resp.,
Rm+l
4. Submanifolds
132
= VUV -g (U,V),,
VUV
(4.7)
U, V
for any vector fields
Rm(1)
tangent to
,m(-1))
(resp.,
In particular, we have
r = 1,...,m-n ,
7x r = °X'r
for any X
tangent to
Rm*1.
totally umbilical in
M
Thus, the submanifold
M.
is also
Hence, we may conclude that
M
is
contained in the intersection of an (n+ 1)-dimensional linear subspace of
3k m+1
we see that
M
and
Rm(1)
(resp.,
3Rm(-1))
.
From this,
is contained in a small hypersphere of an
(n + 1)-dimensional totally geodesic submanifold.
Remark 4.1.
(Q.E.D.)
Totally umbilical submanifolds in complex-
space-forma and in quaternion-space-forms are classified in Chen-Ogiue (2] and Chen (14], respectively.
For a systematic
study of totally umbilical submanifolds in locally symmetric spaces or in Kaehler manifolds, see Chen (17, Chapter VII]. Let
M
be a submanifold of a Riemannian manifold
If the second fundamental form H
of
M
in
M
and the mean curvature vector
satisfy
g(h(X,Y),H) = fg(X,Y)
(4.7)
for some function umbilical.
h
M.
f
on
M,
then
M
is called pseudo-
As a generalization of Proposition 4.1 we have
the following (Yano and Chen (1)).
.
§4. Totally Umbilical Submanifolds Proposition 4.2.
Let
M be a pseudo-umbilical submanifold
Rm(k)
of the real-space-form
curvature vector, then either
Rm(k)
M has parallel mean
If
.
M
is a minimal submanifold of Rm(k)=
is a minimal submanifold of a small hypersphere of
M
or
F1,
.
Proof.
Let
M
be a pseudo-umbilical submanifold of Then the mean curvature
with parallel mean curvature.
Rm(k)
M
a = 0,
is minimal in
a = (HJ
is constant.
If
a
is non-zero.
Then the unit vector
that
of
is parallel,i.e.,
H
133
in the direction
If Rm(k) = Rm ,
DP = 0.
Assume
M.
we
consider the vector field
y(p) = x(p) +
(4.8)
where
is the position vector of
x
any tangent vector on
p
Sa
M
constant.
Rm
This shows that
centered at
y = c
the mean curvature vector and
H
g
M
y
is
lies in the hypersphere
S
of
and with radius H
is always perpendicular to S.
be
Thus
of
M
in
is parallel to the radius vector
the hypersphere
X
AgX
is pseudo-umbilical, we find AS = al.
M
Let
We have
M.
Xy = v x+1a XS = X-a Since
Rm.
in
S.
Thus,
Now, because
a-1.
Rm
is parallel to
x _c,
M
we find that
is minimal in
4. Submanifolds
134
If
of
k ¢ 0.
Rm+1
the result.
we just regard
defined by (4.1).
Rm(k)
as the hypersurface
Then a similar argument yields (Q.E.D.)
§ 5. Minirnal Submanifolds
Minimal Submanifolds
§5.
Let x : M - Rm
be an isometric immersion of an M
n-dimensional Riemannian manifold
a fixed point M
in
Rm
p
in
M.
Let
Rm.
into
M
be an orthonormal local frame on
of
135
ell...len
ve ei = 0
such that
at
denote the position vector
x
Then we f ind
.
n (Ax)
Let
(e) (ex)
P
i=1
1P
(v
1
ei e) 1P
n
i=1
h(ei,ei)p = -nHp
Hence, we have the following well-known results.
Lemma 5.1.
Let x : M -+ Rm be an isometric immersion.
Then
Ax = -nH
(5.1)
Corollary 5.1.
.
x : M -4 Rm is a minimal immersion if
and only if each coordinate function
xA
of
x = (x1,...,xm)
is harmonic.
This corollary follows immediately from Lemma 5.1.
Since every harmonic function on a compact Riemannian manifold is constant (Corollary 2.1.2), Corollary 5.1 implies Corollary 5.2.
of
Rm .
There are no compact minimal submanifolds
4. Subnwnifolds
136
Proposition 5.1. (Takahashi [1)). be an isometric immersion.
(1)
).>0,
(2)
x(M) c So-1(r),
of Rm
Ax = Xx,
If
x :M 4 So-1(r)
(3)
Furthermore, if
X ¢ O,
where So 1(r)
centered at the origin
0
x :M + Rm
Let
then
is a hypersphere
and with radius
r
is minimal.
x :M + So-1(r)
is minimal, then
Ax = (n/r2)x. Proof.
If
Let
H= -()./n)x.
Ax = Xx,
X
be a vector field tangent to
<x,X> = 0
(5.2)
Thus
then by Lemma 5.1 we have
31 0,
.
.
X<x,x> = 2<x,vxx> = 2<x,X> = O.
constant on
Therefore
This proves that IxI
M.
Let
forms of
in
M
respectively.
and
h, h'
Fm ,
M
Rm
of
is
Thus,
M
centered at
be the second fundamental
Ti
in So 1(r), and So-1(r)
in
h(X,Y) = h'(X,Y)+I(X,Y).
Then we have
the mean curvature vectors
<x,x>
is constant.
is immersed into a hypersphere So 1(r) the origin.
we have
M,
H, H'
M
of
in
Rm
Rm Thus,
and So1(r)
n
where
n E 1i(e.,e.),
satisfies
H = H'+ H,
e1,....en
an orthonormal frame of
perpendicular to this implies that Because
X
So-1(r) H' = O.
at
H
i=1
and
p
M
Thus
is an eigenvalue of
M.
a
Since
Hp
x(p)
M,
is
is parallel to
is minimal in on
and
X > O.
x(p),
So-1(r). Now,
§ S. Minimal Submanifolds
137
n
ei'x-rPZ>(xZ) r
nHp = E <' e i=1 i
Thus
= - 2 E <ei.ei>x(p)
r
This proves (1) and (2).
), = 2 .
The last statement
r
is clear.
(Q.E.D.)
Proposition 5.1 shows that minimal submanifolds of spheres are given by proper functions associated with a nonzero eigenvalue
of
a
For a compact symmetry space
A.
a nonzero eigenvalue
a
of
on
a
M,
M
and
we may indeed construct
such a minimal immersion as follows:
M = G/K
Let
be a compact symmetric space where
a compact connected subgroup of of
K
G.
Assume that
is irreducible.
metric on
M
(such
M Let
> ).
of
Let
X.
eigen-space
(5.3)
K
<
<
,
a closed subgroup
>
be a G-invariant Riemannian
>
is unique up to scalar multiple Let
a
be the Laplacian of
For each ) we denote by
m.&
the multiplicity
be an orthonormal basis of the V),
(with respect to
(
)).
xX : M 4 RR\
by
(5.4)
is
is orientable and the isotropy action of
and thus naturally reductive). (M. <
and
i(M)
G
x(p) = 2 ($1(p).....4 mx
(p))
We define a map
4. Submanifolds
138
Then
S0
defines an isometric immersion of
xIL
(M,c<
into
>)
,
1
for some
(1)
(Indeed E
c > 0.
T(M);
nonzero bilinear form on
thus X d4i 0 d4i = c<
>
,
Now, applying Takahashi's result, we conclude
for some
c > 0).
that
is a minimal immersion and
xIL
is an invariant
d4i
c =
these as the following well-known result.
n.
We summarize
(Takahashi (1),
Wallach [1)). Theorem 5.1.
Let
M = G/K be an irreducible compact
symmetric space equipped with a G-invariant Riemannian metric <
Then for any nonzero eigenvalue
>.
A
of
&
there is an isometric minimal immersion of
(M,<
=
1
into a hypersphere S'
x4l
of
on
(r)
of R
where
r
is the i-th nonzero eigenvalue of
If
). i
of
M = G/K
A.
r
M
/
then
is sometime called the i-th standard immersion
M.
Example 5.1.
Let S2(r) = ((x,y,z) E R3 Ix2+y2+z2
=
r2}.
Then, according to Proposition 2.5.3. we know that the eigen-space kk
of
p)
Vk
(associated with the k-th nonzero eigen-value
is given by
polynomials of degree
11k,
k
on
the space of harmonic homogeneous
R3
restricted to
From this, we see that the standard immersion of R 3
is the first standard imbedding of
S2(k). S2(1)
S2(1).
We consider the following homogeneous polynomials of degree
2;
in
§5. Minimal Submanifolds
u1=yz,
139
u3=xy,
u2=xz,
(5.5) U
4 =
z(x2-y2)
It can be verified that their restrictions to V2 = V2.
are harmonic on
u1,...,u5
of
x2
S2(1)
into
(5.5) gives a minimal isometric immersion of S4(
and
defined by
3R5
S2(1)
It is the second standard immersion of
).
R3
form an orthonormal basis of
S2(1)
Thus, the map
6 (x2+y2-2z2)
u5 =
,
into
S2(1)
and
3
it also gives the first standard imbedding of
R P2
into
R5
Similarly, the following homogeneous polynomials of degree 3;
ul =
z(-3x2 - 3y2+ 2z2)
1
u2 = 24 x(-x2 -y2+4z2) ,
2
u
=
15 z(x2-Y 2
u4 =
12
3
24 x(x2
- 3y2)
(5.6)
5= u7 =
24 Y(-x2 - y2 + 4z2 )
u6 = 116 xyz
24 y(3x2-Y2)
are harmonic and their restriction to basis of
A(3.
The map
x3
S2(1)
of
a minimal isometric imbedding. of
,
S2(1)
form an orthonormal
into S6(1) c R7
is
It is the third standard imbedding
S2(1).
The k-th standard immersion space
M
is an imbedding if
M
xk
of a rank one symmetric
is different from a sphere
4. Submanifolds
140
In the case of the k-th standard immersion
or
k
of
Sn with even
map of
is odd.
R Pn.
k,
the immersion is a two-sheet covering
§ 6. The First Standard Imbeddings of Projective Spaces
141
The First Standard Imbeddings of Projective Spaces
06.
In this section we will construct the first standard imbedding of a compact symmetric space of rank one.
Such
imbedding had been considered in various places. (cf. Tai [11, Little [2), Sakamoto (1), Ros (1), Chen [24)). Throughout this section,
of real numbers, the field Q
z
of
will denote the field R
of complex numbers or the field
R c C c Q.
In a natural way,
of quaternions.
element
C
F
we define the conjugate of
F ,
z
For each as follows:
If
z = z0+ z1i+ z2j+ z3k E Q z0,z1,z2,z3 E R ,
with
,
then
z = z0-z1i-z3j-z3k is in
If
z
of
z.
If
C,
z
is in
z
coincides with the ordinary complex conjugate
R,
z = z.
It is convenient to define
2
if F = R , if F = C
4
if
1
d = d(F) =
For a matrix transpose of
A
A
over
F ,
F = Q
denote by
and the conjugate of
A,
At
and A
respectively.
the
4. Submanifolds
142
be a column vector.
z = (zi) E Fm+ 1
Let
operates on
A = (aij), 0 < i, j S m;
.
/a00
.
A matrix
by the rule:
z
aOm
z0
Az =
(6.1)
\a
MO
We will use the following notations:
M(m+ l;F) = the space of all (m+ 1) x (m+ 1) matrices over
F
,
H(m+ 1;F) = (A E M(m+ 1;F)
A*
= A) _
the space of all (m+ 1) x (m+ 1) Hermitian matrices over F , U(m+ 1;F) = (A E M(m+ 1;F) I A*A = I) where
A* = A and
I
is the identity matrix.
If
then A is a symmetric matrix. Moreover, U(m+ 1;R) = 0(m+ 1), U(m+ 1;(C) = U(m+ 1), and U(m+ 1;Q) _ A E H(m+ 1;3R)
Sp(m+ 1). Fm+l space over
can be considered as an (m +1)d-dimensional vector
R
(6.2)
And M(m+ 1;F)
with the usual Euclidean inner product:
.
can be considered as an
(m+ 1)2d-dimensional
Euclidean space with the inner product given by
§ 6. The First Standard Imbeddings of Projective Spaces Re tr(AB*)
(6.3)
143
2
If
A, B belong to H(m+ 1;F) , we have
(6.4)
tr(AB)
Let
F Pm
denote the projective space over
F Pm
F.
is
considered as the quotient space of the unit hypersphere S(mtl)d-1 z
with
that
where
zX,
lx
Fm+l
= (z E
*
1z z = 1]
obtained by identifying
is a column vector and
z
The canonical metric
= 1.
invariant metric such that the fibering is a Riemannian submersion. R Pm
is
1,
on
go
is the -S(mFl)d-1
. F Pm
the holomorphic sectional curvature of QPm QPm
is
(zi)
0
the point in
with
zp = 1,
F Pm
acting on
U(m + 1;F)
with the homogeneous
zl = .. = zm = 0.
Then
the isotropy subgroup at 0 is U (1: F) x U (m; F) . Thus we have the following well-known isometry: (6.5)
µ : F Pm y U(m+ 1;F)/t)(1;F) xU(m;F)
The metric on the right is
Define a mapping
U(m+ 1;F)-invariant.
: S (m+l)d-1 m
-s H(m+ 1;F)
4,
4.
is
Such an action induces an action of U(m+ 1;F)
Denote by
coordinates
such
Thus, the sectional curvature of
Using (6.1), we have an action of
F Pm.
F Pm
n
and the quaternion sectional curvature of
S(m+l)d-1
). E F
as follows
on
4. Submanifolds
144
2
Z0Z1
IZ0I
m(Z) = zz*
.
.
.
.
m 0
z = (zi) E
S(m+l)d-1
induces a mapping of
.
.
.
We simply denote cp('rr(z))
.
.
.
.
Z0Zm
.
.
.
.
.
.
Izml
Then it is easy to verify that
F Pm
into H(m+ 1;F) :
m(7r(z)) = cp(z) = zz
(6.7)
.
2
zm21
z z
for
.
.
by
V(z)
*
if there is no confusion.
Define a hyperplane H1(m+ 1;F) by H1(m+ 1;F) = (A E H(m+ 1;F) tr A = 1). Then we have dim Hl(m+1;F) = m+m(m+1)d/2. F Pm under
From (6.6), we can prove that the image of
cp
is
given by
cp (F m) = (A E H(m+ 1;F) I A2 = A and tr A = 1)
(6.8)
Let U(m+ 1; F)
act on
by
M(m+ 1;]F)
P(A) = PAP 1
(6.9)
for P E U(m+ 1;F)
and A E M(m+ 1;F) . Then we have
(6.10)
Hence, the action of
of M(m+ 1;F) .
(6.11)
U(m+ 1;F)
.
preserves the inner product
Moreover, we also have cp(Pz) = P(cp (z) ) E cp(F Pm)
I
145
§6. The First Standard Imbeddings of Projective Spaces
Thus, we have the following.
for z E F Pm and P E U (m + 1;F ). Lemma 1.
The imbedding
(Tai [1))
cp
F Pm
of
into
given by (6.7) is equivariant with respect to and
H(m + 1;F)
invariant under the action of U(m+ 1;F) . Now, we want to show that the imbedding standard imbedding of Consider a curve
A(t)
F Pm.
Let
in
M with
A
be a point in
A(O) = A
From A2(t) = A(t),
A'(O) = X E TA(F Pm).
Because the dimension of the space of all such that
XA + AX = X
md,
is
is the first
tp
cp(F Pm).
and
we find XA + AX = X.
X
H(m+ 1;F)
in
we obtain
TA(F Pm) = (X E H(m+ 1;F) I XA+AX = X)
(6.12)
There is another expression of
TA(F Pm)
.
given as
follows:
For
u, v E
,
S(m+l)d-1
a point in
identify
IF M+l
v
we define and
T
S(m+l)d-1
be a curve in
Then
A(t) = a(t)a(t)*
Let
under
7r(
with
is a curve in
be
z Tz(S(m'l)d-1).
a vector in
v
and its image in
a(t)
a(u,v) = u v.
a(O) = z tp(F Pm)
and
7r*.
We
Let
a'(O) = v.
through A = zz*.
From this we find
M*(v) = vz* +zv*
.
Therefore, we have
(6.13)
*
*
TA(F Pm) = (vz + zv I v E Fm}
and
a(z,v) = 0)
,
4. Submonifolds
146
S(mfl)d-1
where A = zzz E
A vector if and only if is in
TA(F Pm).
is normal to
in H(m+ I; F )
g
for all
<X,g> = 0
if and only if
TA(F Pm)
X
in
F Pm
TA(F Pm).
at
Thus,
for all
tr(Xg) = 0
A
x
in
Therefore, by (6.12), we obtain
TA (F Pm) = (g E H(m+ 1;F) 1 Ag = gA)
(6.14)
For each
in
A
we have
tp(F Pm)
tr(A-m1 1)2
1 I, A-m+l I> =
2
is imbedded in a hypersphere
Therefore,
F Pm
H(m +1;F )
centered at
Let
X
be a vector in
tangent to F Pm. A(O) = A
that of
Y
and with radius TA(F Pm)
Consider a curve
and
A(t).
to
mIl
A'(O) = X.
Because
A(t)
cp(F Pm)
Y(t)
Y(t) E TA(t)(F Pm),
A(t)Y(t)+ Y(t)A(t) = Y(t)
l))1/2.
a vector field
in
Denote by
of
r = (m/2(m +
Y
and
S(r)
so
the restriction (6.12)
,
from which we find
vXY = Y'(O) = A(VXY)+ (pXY)A+XY+YX ,
(6.16)
where
v
denotes the Riemannian connection of the Euclidean space
H(m+1; F) .
Using (6.12) we have
147
§6. The First Standard Imbeddings of Projective Spaces
AXY = XYA
(6.17)
Thus we find
A (XY + YX) (I - 2A) = -XYA-YXA = (XY + YX) (I - 2A) A
.
Hence, by (6.14), we obtain
(6.18)
(XY+YX)(I-2A) E TA(FPm)
On the other hand, by multiplying
A
to (6.15) from the
right, we get
(XY+YX)A+A(vxY)A = 0
(6.19)
.
Therefore, from (6.8), (6.17) and (6.18), we obtain
(6.20)
2(XY+YX)A+A(VXY) + (VXY)A E TA(F Pm)
Combining (6.16), (6.18) and (6.20), we find
h(X,Y) = (XY+YX)(I-2A)
(6.21)
(6.22) where at
A
VXY = 2(XY+YX)A+A(VXY) + (-vXY)A S and
is the second fundamental form of F Pm in Hfm+ 1;3r) v
the induced connection on
find that the mean curvature vector
H(m+ 1;F)
at
A
is given by
ft
F Pm.
of
F Pm
From (6.21) we in
4. Submanifolds
148
H = m (I - (m+ 1)A)
(6.23)
which is parallel to the radius vector A - m11 I. is imbedded in the hypersphere
F PM
is a standard imbedding of
eigenvalue Since X1
of
X
A.
on F Pm,
p
as a minimal
F Pm
associated with an
From Theorem 5.1 we obtain
). = 2(m+ 1)d.
is exactly the first non-zero eigenvalue
2(m+ 1)d
of
)
Using the result of Takahashi (Proposition 5.1),
submanifold. cp
S(
Thus,
we conclude that
is the first standard
cp
We summarize these results as the following
imbedding.
well-known theorem.
Theorem 6.1.
The isometric imbedding cp :F PM . H(m+ 1;F)
defined by (6.7) is the first standard imbedding of
Pm
Moreover, the second fundamental form
H(m + 1;F) .
mean curvature vector
of
i
F Pm
by (6.21) and (6.23), respectively.
of H(m+ 1;F)
hypersphere
S(r)
with radius
r = [2(m
in
h
F Pm
and the
are given
H(m+ 1;F )
And
into
lies in a
centered at
(Il)I
and
+l))1/2
zzw
Let
X
A -
in TA(F Pm),
a(z,v) = 0
there is a vector
and X = vz*+ zv*.
J
v
in
such that
we put
.
defines the complex structure of
in a similar way.
]Fm+1
If F = a,
we may define the quaternionic structure ep(QPm)
For each vector
cp(F Pm).
ix = viz* - ziv*
(6.24)
Then
be a point in
cp(CPm).
(J1,J2,J3)
Similarly, on
§ 6. The First Standard Imbeddings of Projective Spaces Let
and
X = uz* + zu*
Y = vz* + zv*
be two vectors
TA(CPm), where A = zz*, a(v,z) = a(u,z) = 0, S 2(m}1)-1 Then we have z E in
ix = uiz* - ziu* ,
149
and
JY = viz* - ziv*
Thus, we find zu*vz*
uv*+
(JX)(JY) =
(6.25)
= XY
Consequently, by using (6.21), we obtain
(6.26)
h(JX,JY) = h(X,Y)
A similar formula holds for
X,Y E TA(CPm)
for
QPm
in
.
H(m+ 1;Q).
In the remaining part of this section, we shall study the second fundamental form
of
fi
F Pm
in
H(m + 1;3F)
in more
details.
Let
z0 = (110,...10)t
and AO = zOzO.
Then (6.13)
implies
(6.27)
TA (F Pm) =
b
0
b E Fm
X E TA (F Pm) is a unit 0 takes the following form:
Using (6.4), we see that a vector x
b*
X=
0
vector if and only if
O
150
4. Submanifolds O
b*
b
O
b b = 1
X =
(6.28)
Therefore, by using (6.4), (6.21), and (6.28), we obtain
f-1
0
fi(X,X) = 2
(6.29)
0
bb*
Therefore, we get
llh(X,X)JI = 2
(6.30) for unit vectors X
equivariantly in (6.31)
in
TA (F Pm). 0
H(m+ 1;3F),
flh(X,X)jj = 2,
,
Since F Pm
is imbedded
we obtain
for unit vectors
X E TA(F Pm) ,
where A E m(F Pm) . We need the following.
Lemma 6.2.
Let F Pm be imbedded by
Then the sectional curvature
(6.32)
K
of F Pm
4V
into
satisfies
3(4 + 2f((X,Y))
for orthonormal vectors X, Y tangent to F Pm in Proof.
Since
X, Y
H (m + 1; F)
H i m + 1;F )
are orthonormal, (6.31) implies
§6. The First Standard Imbeddings of Projective Spaces
(6.33)
151
32 =
+
On the other hand, from the equation of Gauss, we find
K(X,Y) _
(6.34)
Combining (6.33) and (6.34), we obtain (6.32).
(Q.E.D.)
By using (6.31), (6.32) and Lemma 6.2, we obtain the following.
Let
Lemma 6.3.
M
be an n-dimensional submanifold of
F Pm which is imbedded by mean curvature vectors in
F Pm
IHI2-
X
X A Y
and if of
and
H'
H (m + 1; F) . of
M
in
(H'I2
=
4(n+2) + 2 3n 3n2
F Pm
Then the
H(m+ 1;F)
and
X IC(ei,e.) 7
ii
e1,...,en form an orthonormal basis of Let
Then
into
satisfy
(6.35) where
H
cp
and
Y
T(M).
be two orthonormal vectors in T(F Pm).
is called totally real if
X i JaY, a = 1,2,3,
when
X 1 JY when F = C
F = Q.
A submanifold
is called totally real if every plane section in
is totally real.
A submanifold M
complex submanifold if
in
J(T(M)) = T(M)
CPm
M T(M)
is called a
and a submanifold M
4. Submanifolds
152
of
QPm
for
is called a quaternionic submanifold if
JJ(T(M)) - T(M)
Complex submanifolds and quaternionic submanifolds
a - 1,2,3.
are also called invariant submanifolds.
A submanifold M of
is regarded as a totally real submanifold and as an invariant
R Pm
submanifold of R Pm
in a trivial way.
From Lemma 6.3, we have the following. Let
Lemma 6.4.
M be an n-dimensional submanifold of
F Pm which is imbedded by
cp
IHI2 2
(6.36)
IH'I2+2 nl
equality holding if and only if Proof.
of F Pm
into H(m+ 1; F) . Then we have
M
is totally real in
F Pm.
It is known that the sectional curvature
is 2 1,
equality holding if and only if
totally real (cf. Chen and Ogiue (1).)
K(X,Y)
X A Y
Thus, by Lemma 6.3, we
obtain Lemma 6.4.
Lemma 6.5.
submanifold of q.
is
(Q.E.D.)
Let
M
be an n-dimensional (n > d)
F Pm which is imbedded into
minimal
H(m + 1;F)
Then we have
(6.37)
IHI2 <
equality holding if and only if invariant submanifold of
F P. m
2(n+d) n
n s 0 (mod d)
and M
is an
153
§ 6. The First Standard Imbeddings of Projective Spaces Proof.
F = R ,
If
F = C,
for any
K(ei,ej) = 1
from the fact that If
this lemma follows immediately i, j = 1,...,n.
then, from formula (2.6.3), we have
K(ei,ej) = 1+ 3<ei,Jej>2
(6.38)
Thus, by Lemma 6.3, we find 2 n+l) +
IHI 2 =
(6.39)
Denote by
n
n 2
n2 i,j=l
the endomorphism of
P
for
i
Jej>2
defined by
T(M)
Then, by (6.39), we find
X, Y E T(M).
IH12=2nn1
(6.40)
<e
F,
+ 2IIP112 n
Since have
n
is nothing but the tangential component of
P
with the equality holding if and only if
IIPII2 S n,
is even and
T(M)
is invariant under
a complex submanifold of 2(n+2),
1H12
n
given by (2.6.5). M,
M
is
M
is
CPm R
takes the form
Thus, for any orthonormal vectors
we have 3
(6.41)
i.e.,
Thus, we find that
the curvature tensor
F = Q,
tangent to
QPm
J,
with equality holding if and only if
a complex submanifold of If
JITM,
K(X,Y) = 1+ 3 E <X,JrY>2 r=l
Thus, by combining (6.34) and (6.41), we find
X, Y
we
4. Submanifolds
154
2(n+1)
=
(6.42)
+
n
3
-L
IH:1
E <ei,J e.2
F,
r j>
n 2 r=1 i,]=1
Define the endomorphism for
IH12
Since
nl
2
=
+ n2
holding if and only if
IH12 < 2(n+4)/n, equality
is a multiple of
n
quaternionic submanifold of
A E cp(CPm)
fundamental form
(11P1112+ 11P2112+ 11P3112)
(6.43) implies
IIPr112 < n,
Let
by
we have
X, Y E T(M),
(6.43)
Tp(M)
of
Pr
h
of
M
and
CPm
H(m + 1;C)
in
is a
(Q.E.D. )
QPm.
X,Y,Z E TA(CPm).
and
4
Then the second satisfies
Thus, we find
h(JX,JY) = h(X,Y).
(vXfi) (JY,JZ) = DXh(Y,Z) - Fi(vxY,Z) - Fi(Y,vxZ) (vXFi) (Y,Z) (v,I)(Y,JY) = 0.
from which we find
we obtain
(vYh)(X,JY) = 0.
In particular, this implies
(V c)(Y,Y) = (vlri)(JY,JY) = 0.
H(m+ 1;C) is,
h
Since the ambient space vh = 0,
is Euclidean, this implies that
that
is parallel.
Remark 6.1.
It was proved in Little (2) and Sakamoto [1]
that the second fundamental form
H(m+ 1;F) , F = R , C or (6.44)
Applying Codazzi equation,
Q,
Fi
under
vh = 0
.
of each cp
F Pm
in
is parallel, that is,
155
§6. The Fast Standard Imbeddings of Projective Spaces
One may obtain the first standard imbedding
Remark 6.2.
of the Cayley plane
OP2
in a similar way as follows:
Cay denote the Cayley algebra over
(e0 = l,e1,...,e7)
Let
I R.
be the usual basis for Cay. For a z = z0 +Z 1e 1 + in Cay, the conjugate of
z
Cay = R8
and the norm of
is defined as
x
H(3;Cay) be the space of Then
x
for
<x,y> = Re(xy) = E xiyi
. + z7 e7
is defined by
z = z0-zle1-... -z7e7 The usual inner product in
Let
.
is given by xiei,
Y = E Yiei
jxI =
<x,x>1"2.
Let
Hermitian matrices over Cay.
3 x3
H(3;Cay) is a Jordan algebra under the multiplication:
A *B = Z(AB+ BA), Define an inner product in
tr(AB)
for A,B E H(3;Cay) H(3;Cay) = R 27
for
by
A,B E H(3;Cay)
2 Let
H1(3;Cay) = (A E H(3;Cay) Itr A = 1).
projective plane
OP2
is defined as
OP2 = (A E H1(3;Cay) I
The
OP2
Then the Cayley
A2
= A)
.
with its induced metric becomes a compact rank-one
symmetric space with maximal sectional curvature 4. it is known that
OP2
Moreover,
is a minimal 16-dimensional submanifold
4. Submanifolds
156
S23(--)
of a hypersphere
with radius
in
H1(3;Cay).
-L (see, Tai (1), Little [2), and Sakamoto [11). OP2
see that the mean curvature IHl of
(Hl =
(6.45) Moreover, if
M
H
of M
then the
,
equality holding if and only if n at
OP2.
H1(3;Cay) satisfies
in
JHI S 2
(6.46)
H1(3;Cay) satisfies
.
is a minimal submanifold of
mean curvature vector
TA(M)
in
From this, we
8, and each tangent space
is a subspace of a Cayley 8-plane of
A E M
TA(OP2).
Furthermore, one may prove that the first nonzero eigenvalue k1
OP
of Laplacian and the volume of
vol(OP2)
(6.48)
(6.49)
are given respectively by
11 = 48 ,
(6.47)
where w
2
7
=
7T, w
is given by
w=
r fr sin8(y -X) I sin 2(y-x) I7 2
0
x
dy dx
§7. Total Absolute Curvature of Chern and Lashof Total Absolute Curvature of Chern and Lashof
{7.
Let
be a closed oriented curve in the 2-plane
C
As a point moves along 0
or rotate
2nrr
3R2
the line through a fixed point
C,
and parallel to the tangent line of
an angle n
157
times about
n
is called the rotation index of
C.
rotates through
C
This integer
0.
If
C
is a simple curve,
n = *1.
Two closed curves are called regularly homotopic if one can be deformed to the other through a family of closed smooth Because the rotation index is an integer and it
curves.
varies continuously through the deformation, it must keep Therefore, two closed smooth curves have the same
constant.
A theorem of
rotation index if they are regularly homotopic.
Graustein and Whitney says that the converse of this is also Thus, the only invariant of a regular homotopy class
true.
is the rotation index. Let
closed curve length
(7.1)
where
be the Euclidean coordinates of the
(x(s),y(s))
s.
C
in
R2
which is parameterized by its arc
Then we have Y"(s)
x"(s) = -x(s)Y'(s) ,
x(s)
denotes the curvature of
- x(s)x'(s)
C.
Let
angle between the tangent line and the x-axis.
de =
xxy2+ x1# x m y !
Y
2
ds = x ds
8(s)
be the
We have
4. Submanifolds
158
From this we obtain the following formula:
(7.2)
J
n = the rotation index
x ds = 2nrr,
.
C
Using (7.2) we may conclude that the total absolute curvature, of
f InIds,
C
satisfies
Ix Ids > 2n
(7.3) C
The equality holds if and only if
C
is a convex plane curve.
This result was generalized by W. Fenchel [1] in 1929 to closed curves in
R3
and by K. Borsuk [1] in 1947 to closed
Rm, m > 3.
curves in
In 1949 - 1950, Fary [1] and Milnor [1]
obtained the following improvement to knotted curves. Theorem 7.1.
If
Rm
is a knotted closed curve in
C
,
then
(7.4)
S
Ix Ids > 4rr
.
C
The Fenchel-Borsuk result was extended by S.S. Chern and R.K. Lashof (1] in 1957 to arbitrary compact submanifolds in
Rm Let
which we will discuss as follows. x : M - Rm
be an isometric immersion of an
n-dimensional closed manifold
The normal bundle
T1(M)
of
M M
into a Euclidean m-space. in
Rm
is an (m - n) -
dimensional vector bundle over M whose bundle space is the subspace of
M X Rm,
consisting of all points
(p,g)
so
§ 7. Total Absolute Curvature of Chern and Lashof and
p E M
that
M
is a normal vector of
g
respect to the induced metric from Rm
at
With
p.
the normal bundle
is a Riemannian (m-n) - plane bundle over
rI(M)
159
M.
Let
B1
denote the subbundle of the normal bundle whose bundle space consists of all points and
(p,g)
is a unit normal vector at
C
M
of (m-n-1) -spheres over
of degree
to a fiber
m -n- 1
unit normal vectors at of
B1.
is a bundle
B1
on
Then
p.
We denote it by
dVB
.
denote
Then there is a differential
M.
B1
dV
Let
such that its restriction
is the volume element of the sphere
Sp
p E M
and is a Riemannian manifold of
the (Riemannian) volume element of do
Then
p.
endowed with the induced metric.
dimension m -1
form
such that
T'(M)
in
dV A do
SP
of
is the volume element
In fact, this can be seen as
1
follows:
Rm
Suppose that in
Rm
we mean a point
with that of Rm M.
Moreover,
Rm.
Then
Denote by
structure group
in
Rm
p, el,...,em
and an ordered set of
F(Rm)
the space of all frames
is a manifold of dimension
F(Rm)
F(Rm)
p
By a frame
e1,....em whose orientation consistent
orthonormal vectors
in
is oriented.
is a fibre bundle over SO(m).
Rm
m(m+ 1).
with 2the
In what follows it is convenient to
agree to the following range of indices:
1 S i,j,k S n;
Let
wA
n+ 1 S r,s,t S m;
denote the dual 1-forms of
connection 1-forms defined by
1 S A,B,C S m.
eA.
Let
W_A
be the
4. Submanifolds
160
V eB = E WAB eA
(7.5)
Then
-A, -A
satisfy the following structural equations of
Cartan:
dwA
(7.6)
= - E wB n wB
4A = - E wA A wB , B B C
( 7 .7 )
WA + W-B
B
A= 0
Throughout this section, we shall consider
F(Rm)
forms defined on Let
x : M .4 Rm
be an isometric immersion of an
with its image under
t
Rm.
We identify a Let
x,,.
the bundle whose bundle space is the set of
projection B,
M and
B a M
en+l,...,em
is denoted by i.
denote
B
M xF( Rm)
consisting of (p,x(p),el;...,en'en+l" ..,em) are tangent to
as
in a natural way.
n-dimensional Riemannian manifold into tangent vector
WA, WB
such that
are normal to
M.
el,...,en
The
We define the map
by $l(P,x(P),el,...,em) _ (P,em)
(7.8)
Consider the maps
B l--
(7.9) where
i
M x F(Rm) -2-> F(Rm)
is the inclusion and
second factor.
Put
k
is the projection onto the
§7. Total Absolute Curvature of Chern and Lashof (7.10)
Since
WA =
and
d
n
(),i)*wA
WAB = (li)*w8 *
commute with
(Xi)
dwA = - E W
(7.11)
From the definition of
wl...,Wn
B
M,
it follows that
dV = w 1
wr = 0
and
If we restrict these
then the volume element
(7.13)
dV
of
M
is given by
n ... nwn
Moreover, the volume element of
(7.14)
(7.6) and (7.7) imply
,
n W
are linearly independent.
n 1-forms to
161
B1
dVndo = wln ...
is
nu,nnwn+ln...
nwm-1
being equal to the product w n+1 n This is n wm-1' the (m - n - 1) -form on B1 which we are looking for.
do
Since
wr = 0,
(7.11) gives
o=dwr=
wf Awi
Hence by Cartan's Lemma (Proposition 1.3.2), we may write
(7.15)
M
in
wi = E hij W
Rm ,
we have
4. Submanifolds
162
(7.16) where
hid =
<
>
,
,
denotes the Euclidean metric of
Rm
.
Consider the map
v : B1 -+ Sm-1 of
BI
into the unit sphere
Rm
of
defined by
Denote by dj the volume element of
v(p,e) = e.
e = em
Sm-1
Sm-1
is the position vector of
Rm,
in
Sm-l.
Since
(7.5) implies
= win ... A wm_1
(7.17)
Therefore, by (7.15) and (7.17) we find
v c E = G(p,em) W 1 A ... A wn A Wn+ A... A W m_1
G(p,em) = det(h'.)
is called the Lipschitz-Killing curvature at The total absolute curvature
TA(X) -
1
cm-1
where
(7.21)
of the immersion
in the sense of Chern and Lashof, is then defined by
x :M -o Rm,
(7.20)
TA(x)
(p,em).
cm_1
Iv*dZI =
f
1
cm-1
B1
f
G*(p)dV
M
is the volume of unit (m-1) -sphere and
G*(p) = J
IG(p,em)Ida
p
§7. Total Absolute Curvature of Chern and Lashof
163
The famous Chern-Lashof inequality is given by the following.
M
immersion of an n-dimensional compact manifold x
Then the total absolute curvature of
be an
Let x : M - Mm
Theorem 7 .1 (Chern-Lashof (1, 21) .
into
1R
M
satisfies the following
inequality: TA(x) 2 b(M)
(7.22)
Proof.
For each unit vector
height function
a
is a unit normal vector at
g
in
in the direction
ha
a
Sm-1 we define the by
p E M .
ha(p) =
If
.
i.e.,
p,
dhS(p) =
Hence,
p
hg.
is a critical point of the height function
then
dha(P) =
Thus,
then
.
is a critical point of the height function
p
Conversely, if ha,
(p,g) E Bl,
a
is a unit vector normal to
(p,a) E B1.
points of
at
p,
i.e.,
Consequently, we see that the number of all critical
ha which is denoted by
number of points in
we obtain
M
.
M with
a
$(ha)
is equal to the
as its normal vector.
Hence,
4. Submanifolds
164
= Iv*
JB
SaESr-1
1
a
Since for each point if and only if
in a
ha
Sm-l,
has degenerate critical
is a critical value of the map
By Theorem 1.6.1 of Sard the image of the set of
v :B1 -4 Sm-l.
critical points of almot all
13 (ha)c .
I
a
in
v
Sm-1,
has measure zero in ha
Thus, for
Sm-1
is a non-degenerate function.
Therefore,
g(ha)
is well-defined and is finite for almost
all
Sm-1.
By applying Theorem 1.6.2 of Morse we
a
in
obtain (7.22).
(Q.E.D.)
Theorem 7.2
(Chern-Lashof [1)).
Under the hypothesis of
Theorem 7.1, if
TA(x) < 3
(7.23)
then M
is homeomorphic to an n-sphere.
Proof.
Suppose that (7.23) holds.
set of positive measure on
Sm-1
such that if
vector in this set, the height function critical points.
a
critical points.
ha
a
is a unit
has exactly two
Since, by Sard's theorem, the image of the
set of critical points under a unit vector
Then there exists a
such that
v
ha
is of measure zero, there is has exactly two non-degenerate
Applying Theorem 1.6.3 of Reeb,
homeomorphic to an n-sphere.
M
is
(Q.E.D.)
§7. Total Absolute Curvature of (here and Lashof For a hypersurface
of 2
into two parts, then
Rn+l
in
Tp(M)
the tangent plane
p E M,
M
M
M
at
p
does not separate
M with total absolute curvature
we have the following. Theorem 7.3 (Chern-Lashof (11).
of Theorem 7.1, if subspace in
if for each point
is called a convex hypersurface
For an immersion of
.
Rn+1
165
Rn+l
Rn+l
of
TA(x) = 2,
Rm
then
Under the hypothesis
M belongs to a linear
and is imbedded as a convex hypersurface
The converse of this is also true.
For the proof of this theorem, see Chern and Lashof [1). If
dim M = 1,
Chern-Lashof's results reduce to the famous
result of Fenchel-Borsuk.
The Chern-Lashof results also gave
birth to the important notion of tight immersion which serves as a natural generalization of convexity. If
M in
into
R3
x : M -. R3
R 3.
is an immersion of a compact surface
then the Lipschitz-Killing curvature of
reduces to the Gauss curvature
G(p,e3)
(7.24)
G,
M
i.e.,
G(p)
Thus, by (7.22), we obtain the following inequality of Chern and Lashof:
(7.25) where
J
X(M)
jGjdV
Z 2tr(4 - )((M)) ,
M
denotes the Euler characteristic of
M.
166
4.
Analogous to Fary-Milnor's result on knotted curves, R. Langevin and H. Rosenburg [1) obtained in 1976 the following result on knotted tori: Theorem 7.4.
T
Let
(7.26)
f
be a knotted torus in
Iii
.
Then
IGIdV > 16a
T
Recently, N.H. Kuiper and W.H. Meeks [1) improved (7.26) to the following. Theorem 7.5.
(7.27)
Let
T
f
be a knotted torus in
3t3
.
Then
IGIdV > 161r
T
For the proof of this theorem, see Kuiper and Meeks [1).
§8. Riemannian Submersions
167
Riemannian Submersions
§8.
In this section, we will study Riemannian submersions in more detail.
The fundamental geometry of submersions has been
discussed by B. O'Neill (1]. Let
vector YX
be a Riemannian submersion.
7r :M - B
X
of
can be decomposed as
X
M,
is vertical and kX
is horizontal.
the Riemannian connections of X
each tangent vector field horizontal vector field X
and
Y
X
on
and
VX +}1X,
Let B,
v
K(vXY) = vXY
B,
Let
and
v
be
To
there corresponds a unique
B
on M such that
This can be seen as follows:
where
respectively.
7r*X = X.
are any two tangent vector fields of
(8.1)
field of
M
For a tangent
Z
B,
If
we have
.
be any tangent vector
we have
2
+ Y
= X
- <X, [Y,ZI> o 7r +
M,
then
Furthermore, if
V
is a vertical vector
4. Submanifolds
168
7,t [%,V] = [n*]C,ir V] = 0
%,(VXV) = W(VvX)
We define on M
(8.4)
we put
M,
AXY = YV
This definition shows that
CY + *(VUXYY
AX
horizontal and vertical subspaces.
A X Y = -A X
X
If 2
and
Y
N
there is a submersion
X
is vertical,
is an n-dimensional submanifold of ir:M a B.
7r -.N -. N'
where
-1
N'
is a submanifold
commutes and the immersion
M
1n
N
B
f
M
That is, suppose that
such that the diagram
Nf)
fibers.
Moreover, if
are horizontal fields, then
which respects the submersion
B
M and it reverses the
Y[X,Y].
Suppose that
of
.
is a skew-symmetric linear
operator on the tangent space of
AXY = 0.
X, Y be tangent
Let
tensor of the submersion as follows: vector fields on
.
of type (1,2) called the fundamental
A
a tensor
.
is a diffeomorphism on the
§8. Riemannian Submersions
169
We shall now relate the second fundamental forms of the submanifolds
N
and
The discussion will be local, and
N'.
so for convenience we shall consider
M
and
If
we denote by
X
and
imbedded in
N'
respectively with the usual identification of tangent
M',
vectors.
N
X
is a tangent vector of and
XT
respectively the projections of
XN
on the tangent and normal spaces of
the normal space is always horizontal. N
vector fields of
N').
(or
M at a point p E N,
and second fundamental form of
at
N Let
Note that
p.
X. Y
be tangent
Then the Riemannian connection (or N') are given respectively
N
by
,
(or
vXY = (vxY)T)
h(X,Y) _ (vxY)N ,
(or
h'(X,Y) = (vxY)N)
vXY = (vXY)T
.
We give the following lemma for later use. Lemma 8.1 (Lawson [1)) if and only if Proof.
N'
N
is a minimal submanifold of
is a minimal submanifold of
For a point
p
in
N,
B.
we choose an orthonormal
about
p.
Let Fl,...,Fn
be local, orthonormal tangent fields on
N'
about
local vertical fields
Denote by
Then N
Fl,...,Fn is minimal in
E1....,Ed
the horizontal lifts of
M
M
if and only if
ir(p).
F1,....Fn.
170
4. Submanifolds
0 = E (O k=1
j This is equivalent to
1
E
Ek k
)
+ E (Vi F.)N j=1
(VF Fj)N =
Fi
j
tr h' = 0
j=1 .
Fj
(VF ) N
j
(Q.E.D.)
§9. Submanifolds of Kaehler Manifolds
Submanifolds of Kaehler Manifolds
§9.
Let
J
171
M
be a Kaehler manifold with complex structure
and Kaehler metric
For each point
g.
p E M,
M be a submanifold of
Let
denote by Xp
subspace of the tangent space
Tp(M),
the maximal holomorphic i.e.,
VP = Tp(M) (1 J(Tp(M))
(9.1)
if the dimension of
.
M
is constant along
!!p
a differentiable distribution X over a generic submanifold of
M.
and
then
M,
The distribution
of
afp
M
p E M,
we denote by Xp
complementary subspace of submanifold, then 1l1
over
M,
in
flp
defines
is called
is called
the holomorphic distribution of the generic submanifold For each point
M.
M.
the orthogonal
Tp(M).
If
M
is a generic
define a differentiable distribution
A(P, p E M,
called the purely real distribution.
For the
general theory of generic submanifolds, see Chen [17,18). It is easy to see that every submanifold of
M
is the closure
of the union of some open generic submanifolds of Let
X
M
tangent to
be a generic submanifold of M,
JX = PX+FX ,
where
and
For a vector
we put
(9.2)
of
R.
M.
PX and
JX, F
FX
are the tangential and normal components
respectively.
Then
P
is an endomorphism of
is a normal-bundle-valued 1-form on
T(M).
T(M)
172
4. Submanifolds We put
(9.3)
a = dim W
Then we have
dim M = 2 a +
dim RV
.
A generic submanifold M
M
of
if its purely real distribution
W'L
is called a CR-submanifold is totally real, i.e.,
(Bejancu [1], Chen [17,20], Blair-Chen [11).
JV II C Tp1(M), P E M.
Since (P
(9.4)
1u)2
= -id.
we have the following inequality:
P 2 , 2a
(9.5)
with equality holding if and only if
M
is a CR-submanifold.
We mention some fundamental properties of CR-submanifolds as follows:
Theorem 9.1 (Chen [20]).
WL
of a CR-submanifold
M
The totally real distribution
of a Kaehler manifold
M
is
Z
and W
completely integrable. Proof.
Let
X
be a vector field in
vector fields in V. Then
$(X,Z) = g(X,JZ) = 0,
denotes the fundamental form of di = 0.
Thus we have
and
I(
M.
Since
M
where
is Kaehlerian,
§ 9. Submanifolds of Kaehler Manifolds
173
0 = d§(X,Z,W) = X§(Z,W) -Z§(X,W)+W§(X,Z)
- §([X,z] ,W) - §(IW,X] ,Z) - 4([Z,WI ,X)
= -g([Z,WI,JX) Because M, [Z,W]
JX
is arbitrary in V
must lie in
u1
distribution
and
is tangent to
[Z,W]
This shows that the totally real
V1.
is involutive.
u1
implies that
.
Hence, Frobenius' theorem
is completely integrable.
(Q.E.D.)
The proof of Theorem 9.1 given above is
Remark 9.1.
simpler than the original proof of the present author done in early 1978.
This simplified proof is essentially given in
Blair and Chen [1], in which the following generalization of Theorem 9.1 was obtained. Theorem 9.2 (Blair and Chen [1]).
manifold with that
M
d§ = §n w
M be a Hermitian
Let
for some 1-form
w.
Then in order
is a CR-submanifold it is necessary that u1
is
completely integrable. Let
manifold
.
M.
be a differentiable distribution on a Riemannian We put 0
(9.6)
h(X,Y) _
for any vector fields
X, Y
the component of
vXY
in
xY)1
9,
where
(vXY)1
denotes
in the orthogonal complementary
4. Submanlfolds
174
distribution
Let
T(M).
in
.D
el,...,er
be an orthonormal
basis of .8, r = dimR. . If we put r
(9.7)
o
H = z E h(ei,ei) i=1
is a well-defined f-valued vector field on
H
Then
m
(up to sign), called the mean-curvature vector of the distribution 0
B.
A distribution 9 on
M
is called minimal if
H = 0
identically.
For the holomorphic distribution X of a CR-submanifold, we have the following general result: Theorem 9.3 (Chen (20)).
of a CR-submanifold
M
The holomorphic distribution
of a Kaehler manifold
M
a!
is a
minimal distribution.
Let X
Proof.
and
Z
be vector fields in U
respectively.
Then we have
(9.8)
g(Z,v.X) = g(Z,vXX) = g(JZ,VXJX)
and
!!j,
= -g(vXJZ,JX) = g(AJZX.JX)
where and
(9.9)
M,
v
and
v
denote the Riemannian connections of
respectively.
M
Thus, we find
g(Z,vix JX) = -g(A2JX,X) = -g(AJZX,JX)
Combining (9.8) and (9.9) we get
g(g X + vjx3X.z) = 0,
from which we conclude that the holomorphic distribution is always a minimal distribution.
J!
(Q.E.D.)
§9. Submanifolds of Kaehler Manifolds
submanifold of a Kaehler manifold
M
Let
Theorem 9.4 (Blair and Chen [11).
175
be a generic
Then the holomorphic
M.
distribution is completely integrable if and only if
h(X,JY) = h(JX,Y)
(9.10)
for X, Y in Proof.
W. If
Al
is integrable, let N be an integral
submanifold.
The second fundamental form
M satisfies
h'(X,Y) = h(X,Y)+ a(X,Y),
second fundamental form of N
N
in
is a Kaehler submanifold of
where
of a
Since V
M.
M.
h'
N
in
is the
is holomorphic,
Thus
h(X,JY) - h(JX,Y) = h(JX,Y) - h(X,JY)
(9.11)
But the left-hand side is normal to is tangent to
M.
M
and the right-hand side
Thus both sides of equation (9.11) vanish
which gives the desired condition.
Conversely, since
J
is parallel with respect to
v,
0 = h (X , JY) - h (JX , Y ) = JvXY - vXJY - JvY{ + v1,JX
= J[X,Y] -vXJY+v1,JX Therefore tangent to
J M
distribution.
.
applied to the tangent vector field and hence
[X,Y]
[X,Y) is
belongs to the holomorphic
Thus the result follows from Frobenius' theorem. (Q.E.D.)
4. Submanifolds
176
Theorem 9.4 implies the following. Corollary 9.1 (Bejancu [1]). of a Kaehler manifold
W
M.
M be a CR-submanifold
Let
Then the holomorphic distribution
is completely integrable if and only if
h(X,JY) = h(JX,Y)
for X, Y in V. In contrast with the integrability of minimality of
and the
W!'`
for CR-submanifold, we have the following
Al
theorem.
Theorem 9.5 (Chen [21]).
of a Kaehler manifold then either Proof. fold
M.
of W. (9.12)
for
Z
A!
M.
If
Let
M
be a compact CR-submanifold
H2k(N) = 0
is not integrable or W
for some
k S dim V ,
is not minimal.
Let M be a compact CR-submanifold of a Kaehler mani-
Choose an orthonormal local frame
Denote by
w1 ,...,w2a
the 2a 1-forms on
w?(Z) = 0;
wI(ei) = bi;
in 0, where
ea+j = Jej.
W=W
(9.13)
ln
dw =
2a 7,
i=1
(-1)1 wl n
M
i,j = 1,...,2a
Then
...
M.
From (9.13), we have
ndwl n
...
nw2a
From a straightforward computation, we can prove that if and only if
satisfying
Aw2a
is a well-defined global 2a-form on
(9.14)
el,...,ea,Jel,...,Jea
dw = 0
§ 9. Submanifolds of Kaehler Manifolds
dw(Z1,Z2,X1,...,X2a-1) = 0
(9.15)
177
,
dw(Z1,X1,....x2a) = 0
(9.16)
Z1, Z2
for any vectors
in
Wl
and
(9.15) holds when and only when Wl
Xl,...,x
But
is integrable and (9.16) holds
is a minimal distribution.
when and only when W
V.
in
2,-l
But for a
CR-submanifold of a Kaehler manifold these two conditions hold automatically (Theorems 9.1 and 9.3), therefore,the 2a-form w
Consequently, we obtain the following.
is closed.
Lemma 9.1. Kaehler manifold
For any compact CR-submanifold the 2a-form
M,
w
of a
M
defines a deRham cohomology
class given by
c(M) _ [w) E H2a(M)
(9.17)
a = dimcl.
where
We need the following. Lemma 9.2.
The cohomology class
compact CR-submanifold non-trivial class if
M Al
c(M) E H2a(M)
of a Kaehler manifold is integrable and
We choose an orthonormal local frame Jea,e2a+11....
X
and
e2a+1
e2a+g
such that
" .. 'e2a+iB
the dual form of
All
is a
is minimal.
e1,...,eaJel,...I
el,...,ea,Jel,...,Jea
are in W. Denote by
el,...,ea,ea+l
M
of a
are in
w1,...,w2a+H
" . . .e2a'e2a+1 " . 'e2a+13'
4. Submanifolds
178
where
Then
ea+i = Jei.
Hodge star operator
(9.18)
*
*w =
Since
w2a+1(X)
(9.18) implies that
Applying the
w = wl A ... n w2a. to
w
we obtain
w,
2a+1
n
2a+$ ...w n
= ... = w2a+o(X) = 0
for all
X
in
Al,
if and only if
d*w = 0
(9.19)
d*w(X1,X2,Z1,....Z8 1) = 0 ,
(9.20)
d*w(Xl,Z1,...,Z13 ) = 0
in
Af
and
(9.19) holds if and only if
K
is integrable; and (9.20) holds
for all vectors
if and only if
and VI
X1, X2
All
(-1)2an+n+1*d*w
= 0,
i.e.,
is a harmonic 2a-form on
M.
w
*w
H2a(M).
Because
ifl.
is integrable
is closed.
Thus,
is also co-closed.
Because
Theorem 3.3.2 of Hodge-deRham implies that a non-trivial class in
in
We see that if V
is minimal.
is minimal, then the 0-form
bw = w
Z1....,Z8
w
Therefore,
is non-zero, c(M) _ [w)
represents
This proves Lemma 9.2.
we choose an orthonormal local frame e
in
M
are in 1 U)
'ea'ea+l'...,eo+8.ea+p+l' ...,em,Je1,...,Jem 1.
in such a way that, restricted to
V
m ,...,w ,W
We put
and
ea+ 1,...,er a
,...,Um
are in
All.
the dual frame of
M, e1....,ea,Jel,....Jea
we denote by ell ..
,em,Jel, ..,Jem.
179
§9. Submanifolds of Kaehler Manifolds
'r 1
9A = wA +
*
wA
8A = wA -
,
'r 1
*
W
.
A = 1,...,m
.
*
Restrict
8A
and
0A.s
to
s
we have
M,
8i = 41 = wl,
for
i = a+ 1,...,a+ is
8r= er=0
for
r=a+13+l,...,m
(9.21)
The fundamental 2-form
M
of
I
is a closed 2-form on M
given by
= E8AA8A A
*
Let
§ =
immersion
i% i
of
M
into
(9.22)
4
a cohomology class
}
via the
j=1
is a closed 2-form on M and it defines [0)
in
H2a(M).
(9.22) implies that the class
(9.23)
induced from
Then (9.21) gives
M.
2
It is clear that
If K
M
be the 2-form on
c(M)
Equations (9.13) and and the class
satisfy
[1)
[,)a = (-1)a(a!)c(M)
is integrable and V 1
(9.13) imply that
H2k(M) ¢ 0
is minimal, then Lemma 9.2 and for
k = 1,2,...,c.
proves Theorem 9.5.
We state the following lemma for later use.
This (Q.E.D.)
180
4. Submanifolds
Lemma 9.3. of
Let
M
be an n-dimensional generic submanifold
CPm which is imbedded in
Then the mean curvature vectors H(m + 1;C)
and
(9.24)
IHI2
CPm
',
H(m+ 1;Q)
H
and
given in (6.7).
cp
M
of
H'
in
satisfy
IH'I2+ 2 (n2+n+2a],
a = dim !
n
equality holding if and only if
m
is a CR-submanifold of
CPm.
This lemma follows immediately from Lemma 6.2, (2.6.3) and (9.5).
Remark 9.2.
of
CPm,
A. Ros [1] proved that if
then
m
is a CR-submanifold
IHI2 = IH' I2+ n (n2+ n+ 2a).
Remark 9.3.
M
If
Kaehler manifold
is a submanifold of a quaternionic
M with quaternionic structure
ill J2' J3'
we put
9p = Tp(M) fl J1(Tp(M)) fl J2(Tp(M)) fl J3(Tp(M)) for of
p E M,
T(M).
then If
-Ap
is the maximal quaternionic subspace
9 : p -..6p
is a differentiable distribution and
its orthogonal complementary distribution totally real, then
M
.8
in
T(M)
is
is called a quaternionic CR-submanifold.
For the general theory of quaternionic CR-submanifolds, see Barros-Chen-Urbano [1].
By using a similar argument as we
give in Lemma 9.3, we also have the following
§9. Submanifolds of Kaehler Manifolds
Lemma 9.4.
Let
181
M be an n-dimensional guaternionic
CR-submanifold of QPm which is imbedded in H(m+ 1;Q) by cp.
Then the mean curvature vectors
QPm
H(m+ 1;Q)
and
(9.25)
IHI2 = 1H,12+ 2
H
and
H'
of
M
in
satisfy
(n2+n+ 12a),
a = dimQ.B
n
Remark 9.4.
For quaternionic version of Theorem 9.5,
see Barros and Urbano [1,2].
TOTAL MEAN CURVATURE
Chapter 5.
fl.
R3
Some Results Concerning Surfaces in For surfaces in
R3
,
the two most important geometric and the mean curvature.
G
invariants are the Gauss curvature
According to Gauss' Theorema Egregium, the Gauss curvature is intrinsic.
The integral of the Gauss curvature gives the famous
Gauss-Bonnet formula:
(1.1)
J
G dV = 2r X(M)
,
M
for a compact Riemannian surface
Moreover, the integral
M.
of the absolute value of the Gauss curvature satisfies the following Chern-Lashof inequality:
(1.2)
fM JGJdV k 2ir(4 - X(M))
.
The idea of integrating the square of the mean curvature instead of the Gauss curvature was discussed at meetings at
Oberwolfach in 1%0 (cf. Willmore [4, p.145).)
The first published
result of this subject appeared in Willmore [1,2) which states as follows;
Theorem 1.1.
M
Let
be a compact surface in
we have
(1.3)
f
IHI2dV -. 4,
M
.
R3.
Then
183
§ 1. Some Results Concerning Surfaces in IR3
sphere in
R3
and
a = IHI
map Ae
.
Clearly, a2-G
Proof.
is an ordinary
M
The equality of (1.3) holds if and only if
and
xl
= (hid).
=
4(xl-x2)2
where
O,
are the eigenvalues of the Weingarten
x2
Thus if we divide the surface M
into
3
regions for which
,H,2dV
(1.4)
J
M
and
G > 0
2$
G S 0,
we have
G dV 2 4r
IHj2dV 2 f
G>O
G>O
where the last inequality is obtained by combining (1.1) and (1.2).
if and only if
R3
.
Moreover, equality of (1.3) holds
This shows (1.3). xl = x2,
i.e.,
By Proposition 4.4.1,
M
M
is totally umbilical in
is an ordinary sphere in
R3
(Q.E.D.)
Analogous to Fary-Milnor's results on knotted curves, the present author obtained in 1971 the following unpublished result on knotted tori by investigating its Gauss map (cf. Willmore
[5l.) Theorem 1.2 (Chen 1971).
R3
.
(1.5)
Let
T
be a knotted torus in
Then
J
1HI2dV > 8ir
M By using a very recent result of Kuiper and Meeks (Theorem 4.7.5), Willmore improves inequality (1.5) in 1982 by replacing the sign by strict inequality.
Willmore's argument
5. Total Mean Curvature
184
goes as follows:
is a knotted torus in
T
If
and Meeks' result implies
f
IGIdV > 16rr.
$
Combining this
T
with the Gauss-Bonnet formula, one obtains
This implies
$
8v.
JH J 2dV
R3 , Kuiper
G dV > 8ir.
G>O
T
R3 , we have the following result of
For tubes in
K. Shiohama and R. Takagi (1) and Willmore [3): M
Let
Theorem 1.3.
be a torus imbedded in
R3
such
that the imbedded surface is the surface generated by carrying a small circle around a closed curve so that the center moves along the curve and the plane of the circle is in the normal plane to the curve at each point, then we have
I
(1.6)
IH
12 dV
2 2,, 2
.
M
The equality sign holds if and only if the imbedded surface is congruent to the anchor ring in
R3
with the Euclidean
coordinates given by
xl = (, a+ a cos u)cos v
,
x2 = (,F2 a+ a cos u) sin v
,
x3 = a sin u where
a
is a positive constant.
Proof.
theorem.
,
Let
Let
C
be the closed curve mentioned in the
x = x(s)
be the position vector field of C
parameterized by the arc length.
Denote by
x
and
7
the
185
§ 1. Some Results Concerning Surfaces In IR'
curvature and torsion of
Let
C.
y denote the position
vector of M
in
(1.7)
y(s,v) = x(s) + c Cos v N+ c sin v B ,
where
N
and
B
R3
Then
.
are the principal normal and binormal of
C.
By a direct computation, we find that the principal curvature of
M in
R3
are given by
kl_ -c'
_
1
x Cos v
k2-xccos v-1
Thus the mean curvature vector satisfies
1- 2xc cos v 2 1 - xc cos v I
HI 12
12c
Thus
_ PpL 2v P JM IH I2 dV = J O JO
1- 2xc cos v 2 12c(1-xc cos v) dv do
it 2c J it - x2c2)-1/2 do
O
where
l
is the length of p
(1.8)
J
M
IHj2dV = 2
C.
Therefore,
it
Ixl
do
0 'xcj 1-x2 c2
?irI InIds>4ir, A
O
by virtue of the fact that, for any real variable takes its maximum value
2
at
x
1 .
42
x.
(1- x )
186
5. Total Mean Curvature If the equality sign of (1.6) holds, inequalities in (1.8)
become equalities. planar curve. of radius
Thus, by Fenchel's result,
Moreover,
2 c.
(2c2)-1/2.
x =
This shows that
M
ring of the type given in the theorem.
Thus,
C
is a convex C
is a circle
is imbedded as an anchor The converse is trivial.
(Q.E.D.) Willmore conjectured that inequality (1.6) holds for all 3
torus in
R- .
Theorems 1.2 and 1.3 shows that Willmore's
conjecture valids either
M
is knotted, or
M
is a tube in
187
§ 2. Total Mean Curvature
Total Mean Curvature
{2.
According to Nash's Theorem, every n-dimensional compact Riemannian manifold can be isometrically imbedded in n(3n + 11).
N =
with
On the other hand, "most" compact Riemannian
-
manifolds cannot be isometrically imbedded in hypersurfaces.
RN
Rn+l
as a
For example, every compact surface with non-
positive Gauss curvature cannot be isometrically imbedded in
R3
.
Furthermore, there are many minimal submanifolds of a
hypersurface of
Rm
which are not hypersurfaces of
Rn+1
Hence, the theory of submanifolds of arbitrary codimensions is far richer than the theory of hypersurfaces, in particular,
than the theory of surfaces in
R3
.
Especially, we will see
that this is the case when one wants to study the theory of total mean curvature and its applications.
The first general result on total mean curvature is given in the following. Theorem 2.1 (Chen [2]). submanifold of
Rm.
Let
M be a compact n-dimensional
Then we have
(2.1)
J
IHind V 2 cn
.
M
The equality holds if and only if
M
is imbedded as an
ordinary n-sphere in a linear (n+ 1)-subspace n > 1
and as a convex plane curve when Proof.
Let
x : M
Rm
Rn+l
when
n = 1.
be an isometric immersion
of a compact n-dimensional submanifold
M
into R. m Let
B
5. Total Mean GLrvature
188
be the bundle space consisting of all frames
(p,x(p),el,...I
such that
are orthonormal
en'en+l" .. ,em)
M
vectors tangent to
at
vectors normal to M
Choose the frame
p.
in
are orthonormal
en+1,...,em
and
p
at
el,...,en,en+l,...,em)
p E M. e1,...,en
such that
B
at
H
to the mean curvature vector find that the mean curvature
IHI
p.
em
(p,x(p),
is parallel
Then we can easily
is given by
IHI = n (hll+ ...+ nn
(2.2) and
r = n+l,...,m-1 ,
(2.3)
where in
1j
On the other hand, for each
=
we can write
Bi,
m
em = s=n+l E
(2.4)
where
8s
(pre) E B1,
cos 8 s e s
denotes the angle between
em
and
es.
For each
we put
trace Ae
K1(p,e) =
(2.5)
n
From (2.2), (2.3), (2.4) and (2.5) we find
m (2.6)
(p,em)
K1(p,em) =
E
s=n+ 1 Hence we obtain
cos 8s K1(P,es) = cos 8mIH(p)I
§ 2. Total Mean Curvature IHInIcosn A
1 K1(p,em)I n dV n do = f
S
(2.7)
189
B1
B1
= (2cm-1/cn) J
m
I dV n do
I HI ndV
.
M Let
be a unit vector in the unit sphere
e
he = <e,x(p)>
the height function
on
is a differentiable function on
h
X, Y
fields
tangent to
Since
h
is continuous on
e
and one minimum, say at q',
e
Since
q
is normal to
Xhe = <e,X>.
h
M,
and M.
q
and
q'
.
At
respectively.
q',
q
Thus, we obtain from (2.8) that
.
give the maximum and minimum of
implies that the Weingarten map
Ae
denote the set of all elements k1(p,e),...,kn(p,e)
he,
(2.9)
is either non-positive
definite or non-negative definite at
eigenvalues
Hence
has at least one maximum
e
YXhe =
(2.9)
U
For any vector
YXhe = <e,V YX + h(Y,X)>
(2.8)
and
we have
M,
Consider
It is clear that
M.
M.
e
Sm-1.
(q.e)
(p,e)
of
Ae
in
and B1
(q'.e).
Let
such that the
have the same sign.
Then from the above discussion we see that the unit sphere Sm-1
is covered by
which is defined by
at least twice under the map
U
v(p,e) = e.
v* dE 2 2c
(2.10) J
This shows that
U
M-1
v : B1 -0
Sm-1
190
S. Total Mean Curvature
Since, on
k1(p,e),...,kn(p,e)
U,
have the same sign, we
find IK1(p,e)In = 1n (k1(p,e)+ ...+ kn(p,e) in
(2.11)
? Jkl(p,e) ...kn(p,e)l = IG(p,e)I
Hence, by using (2.7), (2.10), (2.11) and 14.7.18) we obtain p
(2.12)
J
Cn
IHIndV
2c
M
)
I
M-1
IK1(p,e)in dV Ado B1
c
2 (2cn ) S v*d1?cn M-1 U This proves (2.1).
Now, assume that the equality sign of (2.1)
M
We want to prove that
holds.
is imbedded as an ordinary
Rn+l
hypersphere in a linear subspace
Rm
of
when
n > 1.
This can be proved as follows:
We consider the map
y : B1 . Rm;
(2.13) where
is a sufficient small positive number which gives
c
an immersion of B1
(p,e) -# x(p) +ce
B1
Rm
into
as a hypersurface in
In this way, we may regard
.
Rm.
Moreover, because
<e,dy> _ <e,dx> + c<e,de> = 0, e of
B1
in
3R
m
at
orthonormal basis of dual basis of
(p,e).
Thus
T(p,e)BI'
el,...,em-1.
is in fact a unit normal vector el,...,em-1 Let
wl
form an
,w -1
be the
Then by direct computation, we have
§2. Total Mean Curvature
n wl = wl+ c Z hi. w3
(2.14)
j=1
(2.15) Let
Ae
r = n+ 1,...,m- 1
Wr = cwr ,
B1
be the eigenvalue of the Weingarten
A
kA(p,e), of
191
Rm
in
at
Then, by using (2.14) and (2.15),
(p,e).
we may obtain ki(p,e) ki(p,e) _ 1+cki(p,e)
i = 1,2,
'
,n
,
(2.16)
r = n+l,...m-1 .
kr(p,e)
Let
el,...lem-1
(2.17)
where
be the principal directions of Ae.
wB = kA(p,.e)WA
vem =
wm eA.
Put
A
,
veA =
WA eB.
Taking the exterior
differentiation of both sides of (2.17) we find
(2.18)
C BnWA = C kA;C wCnwA
where we put dkA = (2.19)
kA;C wC.
+
kA wBnwB
Let
WBI'ABC WC
Then (2.17), (2.18) and (2.19) imply
(2.20)
We have
kA.B(p,e) = (k8(p,e) - kA(p,e))i-AA
5. Total Mean Curvature
192
Let
U = {(p,e) E BI lkI(p.e) =
V = B1
= kn(p,e) 30'0]
and
Then (2.20) gives
U.
(2.21)
kA;B(p,e) = 0
for
(p,e) E U
.
If we put
dki(p,e) = E ki:A(p,e)wA then we have (2.22)
ki'j(p,e) = 0
for
(p,e) E U
.
Now, by the assumption, the equality of (2.1) holds.
Thus,
all of the inequalities in (2.11) and (2.12) become equalities. Hence, we have
K1(p,e) = 0
(2.1), we see that Let
U
identically on
is a non-empty open subset of
be a connected component of
U'
know that
w(p) = maxJKI(p,e)l,
e
of
K1(p,e)
on
B1
If
and the fact
see that for each point exists a point
rr(U') / M,
(p,e')
p in
runs over
U'.
K1(p,e) = 0
From This, we find that, for each point
over
p
U',
a :Bl + M
on
V,
rr(U'),
we there
w(p) = JK1(p,e')I.
such that
(p,e')
This is a contradiction.
is a non-empty open subset W
in
then by the continuity
Hence there is an open neighborhood of contained in
M.
(p,e)
where
in the boundary of U'
By
Then, by (2.22), we
U.
is a positive constant function on is the projection.
V = B1- U.
p
of the fibre
such that the principal curvatures
in
Thus in
M,
Sm-n-1
B1
which is
ir(U) = M.
there of
B1
k1(p,e),...,kn(p,e)
§ 2. Total Mean Curvature
are equal for all
is true for all
From this we may conclude that
(p,e) E W.
= kn(p,e)
k1(p,e) _
p
in
for all M,
193
M
in Sp n-1.
e
is totally umbilical in
M
Consequently, by Proposition 4.4.1,
M
n = 1,
Rm
.
is imbedded as an
Rn+1
ordinary hypersphere in a linear subspace If
Since this
when
n > 1.
is imbedded as a convex plane curve by the result
of Fenchel-Borsuk.
The converse of this is trivial.
(Q.E.D.)
An alternative proof of inequality (2.1) was
Remark 2.1.
given in Heintze and Karcher [1).
However, their method does
not yield the equality case.
Some easy consequences of Theorem 2.1 are the following. Corollary 2.1 (Chen [5))
Let
M
be a compact n-dimensional
minimal submanifold of a unit m-sphere M
vol(M) _> cn = vol(Sn)
M
The equality holds if and only if Proof.
of
Then the volume of
satisfies
(2.23)
of
Sm.
Rm+l. M
in
Sm
Regard
Since Rm+l
M
is minimal in
vol(M) =
Theorem 2.1, too.
is a great n-sphere in
Sm.
as the standard unit hypersphere
is equal to one.
This proves (2.23).
.
J
Sm,
the mean curvature
Thus, (2.1) implies
HlndV
cn
M
The remaining part follows easily from (Q.E.D.)
5. Total Mean Curvature
194
Corollary 2.2 (Chen (24]) Let M be a compact n-dimensional minimal submanifold of a real projective m-space R Pm constant sectional curvature
of
Then
1.
c
vol(M)
(2.24)
2 is a R Pn
The equality holds if and only if M
imbedded in
as a totally geodesic submanifold.
R Pm
Proof.
Let M
be a compact n-dimensional minimal submanifold
of a real projective m-space R Pm.
covering map r : Sm -o R Pm. submanifold of
7-1(
of (2.24) holds, then
R Pn
is a
it 1(M)
Then
Sm with vol(n 1(M))
Corollary 2.1 to
Corollary 2.1,
Consider the two-fold
M )
,
is a minimal
2 vol(M).
we obtain (2.24).
Applying
If the equality
vol(n 1(M)) = 2 vol(M) = cn.
7-1 (M)
is a great n-sphere in
imbedded in RP m
Sm.
Thus, by Thus
as a totally geodesic submanifold.
The converse is trivial.
(Q.E.D.)
Corollary 2.3. (Chen (24])
Let M be a compact
n-dimensional (n > 1) minimal submanifold of
CPn with
constant holomorphic sectional curvature
Then
(2.25)
vol(M) N
21r
n - 2k
CPk which is isometrically imbedded in
geodesic complex submanifold.
4.
cn+l
The equality holds if and only if a
M
is even and m CPm
is
as a totally
§ 2. Total Mean Curvature Proof.
195
M be a compact n-dimensional minimal S2m+1 -4 CPm CPm. Consider the Hopf fibration rr :
Let
submanifold of
Denote the r 1(M)
by
Then
M.
M
rr:Fl
submersion with totally geodesic fibres
is a Riemannian
S1.
We consider the
following commutative diagram:
i
Since M
is minimal in
S2m+1
CPm,
Lemma 4.8.1 implies that M
minimal in S21. Thus, by applying 2.1 to
(2.26)
M.
is
we obtain
cml ,
vol(M)
with equality holding if and only if M is a great (n+ 1)-sphere S2mf1. On the other hand, because tr:M + M is a Riemannian in submersion with fiber
S1,
Lemma 2.7.2 gives
vol(M) = 2w vol(M)
(2.27)
.
Combining (2.26) and (2.27), we obtain (2.25). sign of (2.25) holds, then of
S2m+ 1.
with fiber
Since S1,
n
:51 4 M
n = 2k
a great (2k+ 1)-sphere of
M
is a
M
is a Riemannian submersion
is even (Adem (1]). S2r
CPk which imbedded in
complex submanifold.
If the equality is a great (n+ 1)-sphere Sn+l
1.
Thus,
R
is
From this we conclude that
CPm
as a totally geodesic
The converse of this is trivial.
(Q.E.D.)
5. Total Mean Quvature
196
Remark 2.2.
Recently, Roo also obtained a lower bound
of the volume of a compact minimal submanifold of CPm by applying our Theorem 2.1.
However, his estimate is not sharp.
Let QPm be a quaternion
Corollary 2.4. (Chen [241)
projective m-space with maximal sectional curvature 4.
M is a compact minimal submanifold of
M
The equality holds if and only if
QPk
then
QPm,
c2 2n
vol(M)
(2.28)
is a
QPk,
n - 4k;
is imbedded as a totally geodesic submanifold in
Proof. Let is minimal in
QPm,
it-(M) with it :S 4n*3 M
If
QPm.
and
QPm.
Since M
is minimal in S43. Applying Theorem 2.1
we obtain vol(S) - c3 vol(M) = 2ir2 . vol(M) by Lemma 2.7.2. The equality case can be obtained in
Thus, we find (2.28).
the similar way as Corollary 2.3.
Corollary 2.5.
(Chen [24])
with maximal sectional curvature
(Q.E.D.)
Let 4
OP2
and M an n-dimensional
minimal submanifold of
OP2.
(2.29)
vol(M) k cn/2n
Proof.
H(3;O)
Then we have
Regard the Cayley plane
as mentioned in ¢4.6.
be a Cayley plane
OP2 as a submanifold in
Since M
the mean curvature vector of M
in
is minimal in
H(3;O)
Then by using Theorem 2.1, we obtain (2.29).
satisfies
OP2, IHl2 S 2. (Q.E.D.)
197
§ 2. Total Mean Curvature
is sharp if
vol(M)
The estimate of
Remark 2.3.
given in Corollary 2.5
n g 8.
Corollary 2.5 (Chern and Hsiung [1))
compact minimal submanifolds in
There exist no
Rm.
This Corollary follows immediately from Theorem 2.1.
It follows from Theorem 2.1 that the total mean curvature
of a compact n-dimensional submanifold in Rm bounded below by
cn - vol(Sn).
is always
On the other hand, according
to Theorem 4.7.1 of Chern and Lashof, the total absolute curvature is bounded below by the topological invariant Thus, it is natural to ask whether if total mean curvature of
M
in
Rm
The answer to this is no.
large"?
b(M)
b(M).
is large, the
is also "proportionally
This can be seen by using
Lawson's examples of compact minimal surfaces in
In
S3.
Lawson [2), he had constructed a compact imbedded minimal
Mg
surface
of genus
with area less than
R4
as surfaces in than as
87r.
However,
(for an arbitrary
g
0)
in
S3
Thus, if we regard Lawson's examples
8,r.
,
g
they have total mean curvature less b(Mg)
2+ 2g which tends to infinity
tends to infinity.
g
Let
11h112
denote the square of the length of the second
fundamental form
h
of
M
in
Rm.
Then by the Gauss
equation, we have
(2.30)
n(n - 1).r : n2IHI2 _ 11h112 ,
5. Total Mean Curvature
198
(n-l)IIh1I2-n(n-1)T
(2.31)
E (n(hr )2-hiihjj) r,i,j
=
n rEi(j E (hr)2+E E (hr-hr)2 ii r i<j 11 ji
0
Thus we obtain
-(n(n
(2.32)
)IIhII2 S T S (')IIhII2
1)
In the following, a submanifold M
Rm
6-pinched in
is called
-T-
(')IIhII2
In view of Lawson's examples mentioned above,
_> -1.
6
Rm
if we have
(n(nbl )IIhII2 for some
in
we give the following. Proposition 2.1 (Chen (12))
Rm.
submanifold in
M
If
total mean curvature of
M
Let
M be a compact n-dimensional
is 6-pinched in
Rm,
then the
satisfies n
(2.33)
where
fM IHIndV
is defined in §1.6.
b(M)
Let
Proof.
Rm
Let
.
fields of of
M,
M
then
M
Rm
Rm.
be a compact n-dimensional submanifold
en+l,... ,em in
The equality sign of (2.33)
is (n -1)-pinched in
M
holds when and only when
in
(1 1+6)2cn]b(M)
.
If
be local orthonormal normal vector e
e = E cos 8 rer.
is a unit normal vector field Thus,
§ 2. Total Mean Curvature
199
Ar = Ae
Ae = E cos grAr
r
r
Hence, we have
IIAeII2 = E cos or cos 8s trace (ArAs)
(2.34)
r,s
The right-hand side of (2.34) is a quadratic form on cos
Thus, we may choose a local orthonormal
,cos gm.
normal frame
en+l,...,em
such that with respect to this
frame field, we have
(2.35)
IIAeII2
= E (r Cos2Or
(2.36)
B1
be the bundle of unit normal vectors of M
(2.37)
B1
f(p,e) = IIAe112 Since all of
(p,e) E B1.
for
Cm
r
We define a function f on
.
...
C r= I I e 112
Let
Rm
Cn+l
f.,Cos2Or = 1,
by ,
are non-negative and
an inequality of Minkowski implies
n
/p
``2/n
f2 dv1
(2.38) iJ
Cr
in
Sp
//
2
n
/
= {
` Sp
(E Crcos2gr)2 da) //
2 (
S E SCr
where p.
Sp
`
1SSp
Icosn0rlda)
/
is the (m -n -1) -sphere of unit normal vectors at
On the other hand, we have the following identity,
5. Total Mean Curvature
200
ICosnglda = 2cm-1/Cn
(2.39) .
Sp
Thus, by (2.36), (2.38) and (2.39) we find
n
c2cn
(2.40) Let
11h,12
G(p,e)
f2 da
p d
3
m-1 Sp
denote the determinant of
Ae.
Then, by using
a relation between elementary symmetric functions, we have IIAeIIn _
Thus, by using (2.40), we find
nn IG(p,e)I.
c
p
(2.41)
JM
nn
IIhIIndV
f B IG(p,e) IdV n do
2cM-1 1
1
Combining this with Chern-Lashof's inequality, we get nn c
(2.42)
J
IIhIIndV
Now, by the hypothesis,
(2.43)
2
M
M
n b(M)
is 6-pinched in
3tm
.
Thus
IIhII2
T ? n(nsl)-
On the other hand, (2.30) gives n(n -1).r = n2IHI2 - IIhII2. (2.43) implies IHI2 2
(2.44)
Thus,
(1)IIhII2 n
Combining (2.42) and (2.44) we obtain (2.33).
If the equality
sign of (2.33) holds, then the equality sign of (2.39) holds.
From this we may conclude that of a linear subspace (2.30),
M
gn+l
m
with
is imbedded as a hypersurface nIHI2
is (n- 1)-pinched in Rm .
= IIhII2.
Thus, by (Q.E.D.)
§ 2. Total Mean Curvature
M
If
Sm-1
of
201
is a minimal submanifold of a unit hypersphere
Rm
,
then M
is b-pinched in Rm M satisfies
the scalar curvature of
(2.45)
n
T
.
-
-1)(1+6
(
In particular,
M
if and only if
n
Rm
is O-pinched in
if and only if
M
In this case Proposition 2.1 implies
has zero scalar curvature. the following.
Corollary 2.7.
M be a compact n-dimensional
Let
minimal submanifold of
Sm.
M has non-negative scalar
If
curvature, then we have n 2 c vol(M) _,
(2.46)
(n))
2 b(M)
In the 1973 Symposium on Differential Geometry held at Stanford University, the present author proposed the following two problems (See Chen [11)). Let
Problem 2.1.
and
be a compact Riemannian manifold
(M,g)
an isometric immersion of
x : M 4 Rm
can we say about the total mean curvature of Riemannian structure of Let
Problem 2.2. an immersion of
M
of M
(or of
x(M))?
x
What
and the
(M,g)?
M be a compact manifold and
Rm
into
total mean curvature of
into Rm .
M
x
.
x :M -4 Rm
What can we say about the
and the topological structure
5. Total Mean Curvature
202
In this book, we shall give a detailed account of the results on these two problems up to date. Let
Remark 2.4.
:(M,g) .. (N,g)
be a smooth map between
We define its energy by the formula
Riemannian manifolds.
(2.47)
2 f
jI4
Il2 dVM
M
where
dVM denotes the volume element of
the differential of with
E,
and
t*
T(O),
is called the tensor field of
4
(cf., Eells and Sampson (1) and Eells and Lemaire (1]).
4
is an isometric immersion, the energy
volume of
nothing but
where
E(t)
In this case, the tension field
M.
n
If
gives the is
.r(4)
times the mean curvature vector of
n = dim M.
is
The Euler-Lagrange operator associated
4.
denoted by
(M,g);
M
If one defines the total tension of
in
N,
I
by
n i 1J
(2.48)
f
dVM
M
then the total mean curvature
IHIdV
j
is exactly the total
M tension of
4,
whenever
0
is isometric.
203
§ 3. Conformal Invariants
Conformal Invariants
3.
Let P
be an m-dimensional Riemannian manifold and
(M,g)
a positive function on
M.
-* g
Then v*
We put
=P
2
9 .
is a conformal change of metric.
g
the Riemannian connections of
g
and
Denote by
and
v
respectively.
g*,
Then equation (2.5.2) gives
v*i'-vXY =(X log p)Y+ (Y log P)X-g(X,Y)U
(3.2)
U = (dp)$.
where
M
Let g
and
be an n-dimensional submanifold of the metrics on
g*
respectively.
M
induced from
For any normal vector field
and
g g
Denote by
A.
of
g*,
M
in
M
we have
(3.3) for
v"XS-vXS = (X log P)S-(Slogp)X tangent to
X
M.
Thus, by using Weingarten's formula,
we find (3.4)
where (M,g)
(3.5)
DXS -D Xt = (X log P)S
D
and
and
D
*
(M,g ),
denote the normal connections of respectively.
Thus we have
DX = DX + (X log p)I
M
in
5. Total Mean Curvature
204
Consequently, the normal curvature tensors
and
RD
RD
satisfy
RD (X,Y) = RD(X,Y)+ DX((Y log p)I)
(3.6)
+ (X log p)DY -D M log p)I) Y - (Y log p)DX - ([X.Y]log p)I
Therefore, by using the definition of Lie bracket, we obtain
RD (X,Y) = RD(X,Y)
(3.7)
This implies the following (Chen (10]).
Let M be a submanifold of a
Proposition 3.1.
Riemannian manifold
M.
Then the normal curvature tensor
RD
is a conformal invariant. Let
M
in
and
h
and
(M,g)
denote the second fundamental forms of
h
(A,-9*),
respectively.
Then, from (3.2),
we find h*(X,Y) = h(X,Y)+ g(X,Y) UN
(3.8)
where UN
denotes the normal component of
Hence, for any normal vector
(3.9)
Let
S
of
M
in
U
restricted to
M,
we get
9(A9X,Y) = 9(ASX,Y)+ 9(X,Y)9(U,S)
el,...,en
respect to
g.
be principal normal directions of Then
AS with
M.
§ 3. Conformal Invariants
-1 P
e1,...,P
-
they are the principal directions of
kn(g) that of A,
Ag = pAt*
respect to
(3.11)
Moreover,
.
If we denote by
AS
and by
Ag = g(U.g)
is a unit normal vector with
g* = p-lg
,
g
then (3.9) implies
ki(S) = ki(S)+ lg ,
g*,
A
the principal curvatures of
k1(g),...,kn(g)
Since
en
M with respect to
form an orthonormal frame of
(3.10)
205
(3.10) gives
P(ki(g*)- k*(S*)) = ki(S) -k (9) i
Now, let
with respect to
be an orthonormal normal frame
g ,l,...,Sm g.
Then the mean curvature vector
H
is given
by
H = n E (E k.(gr))gr
(3.12)
r
i
We put
(3.13)
Te
n(n--1)
It is easy to see that
Te
E E r i<7 is well-defined.
the extrinsic scalar curvature with respect of constant sectional curvature implies
k,
We call g.
If
Ge (M,g)
the equation of Gauss
is
206
5. Total Mean Curvature
Te = T-k ,
(3.14) where
is the scalar curvature of
T
(M,g).
If
M
is
2-dimensional, the extrinsic scalar curvature relates to the
Gauss curvature
G
by
(3.15)
where
T
R'
respect to
e
= G-R'
denotes the sectional curvature of Tp(M).
(M,g)
with
By using (3.11), (3.12), (3.13) we obtain
the following results (Chen [10)). Proposition 3.2.
Let
M be an n-dimensional submanifold
of a Riemannian manifold
(M,g).
(3.16)
(IH12-Te)g
Then
is invariant under any conformal change of metric.
M
In particular, if
is compact, Proposition 3.2 implies
immediately the following (Chen [10]). Proposition 3.3.
Let
M
be an n-dimensional compact
submanifold of a Riemannian manifold
(M,g).
Then
n
(IHI2-Te)I
(3.17)
J
dV
M
is a conformal invariant. If
following
M
is 2-dimensional, Proposition 3.3 reduces to the
§ 3. Conformal Invariants
Proposition 3.4.
Then
(M,g).
Riemannian manifold
be a compact surface in a
M
Let
207
(IHI2+R')dV
(3.18)
J
is a conformal invariant. Remark 3.1.
For a problem related to Proposition 3.4,
see Ejiri (1].
By assuming the ambient space to be a real-space-form, Proposition 3.4 gives the following Corollaries. Corollary 3.1.
a diffeomorphism of metric on
12m
.
M
Let
Itm
Itm
and
which induced a conformal change of
Then we have
JH12dV = f
$
(3.19)
M For
be a compact surface in
JH012dV
.
O(M)
this Corollary is due to Blaschke [1],
m = 3,
White (1].
Equation (3.19) says that the total mean curvature of a compact surface
M
Corollary 3.2.
in
m
is a conformal invariant.
M
be a compact surface in a complete,
IR
Let
simply connected, real-space-form curvature
-1.
Hm(-1)
Then we have
Hj 2dV Z 47r+ vol(M)
(3.20)
M
of constant sectional
5. Total Mean Curvature
208
M
The equality sign holds if and only if in
is totally umbilical
Hm(-1).
can be obtained from Rm
Hm(-1)
Since
Proof.
a conformal change of the metric on
Rm,
by
Proposition 3.4
implies (IHI2 f
where M
M
- 1)dV = $M
Rm
is the surface in
IHI2dV ,
with the induced metric.
Thus, by Theorem 2.1, we obtain (3.20). (3.20) holds, then,
Rm.
2-sphere in
IH,2dv = 47r.
$
Therefore,
M
If the equality of
Thus,
M
is an ordinary
is totally umbilical in
Since totally umbilicity is a conformal invariant, totally umbilical in
Hm(-l).
M
Rm.
is
The converse is easy to see. (Q.E.D.)
Remark 3.2.
Maeda [1) also obtain Corollary 3.2 by
using a quite different method.
For the standard m-sphere G(Sm)
Sm,
one may consider the group
of conformal diffeomorphisms of
of diffeomorphisms on metric on
Sm.
Thus, if
M
i.e., the group
Sm which induce conformal changes of
In this case, Proposition 3.4 reduces to
(IH12+1)dV $M
Sm,
= J
is minimal in
(M) Sm,
(IH+I2+1)dv, then we have
E G(Sm) .
§ 3. Conformal Invariants
vol(4>(M)) S
J
p E G(Sm)
1)dV = vol(ts),
.
0M)
Let M be a compact
Corollary 3.3 (Li and Yau [1)).
minimal surface of Sm f :M -. Sm.
209
given by an isometric immersion
Then we have
vol(M) = Vc(m,f)
(3.21)
where Vc(m,f) =
sup
,
called the m-conformal
vol(4>(M)),
EG(Sm) volume of
f.
Another consequence of Proposition 3.4 is the following. Corollary 3.4 (Li and Yau (1).)
surface in
Rm
.
be a compact
Then
(3.22)
where
M
Let
IH12dV 2 Vc(m,M)
J
Vc(m,M) = inf Vc(m,4>),
conformal mappings of
M
4>
into
,
runs over all non-degnerate
Sm.
Using the inverse of stereographic projection,
Proof.
one forms a conformal immersion
of
4>
M
into
Sm.
Compositing with a Mobius transformation, one may assume that the area of
4>(M)
is equal to
Vc(m,4>).
From Proposition 3.4
we get
(3.23)
f
M
JH12dV = J
(JH
(M)
12
+ 1)dV > vol(4>(M))
.
S. Total Mean Curvature
210 This implies (3.22).
(Q.E.D.)
In the remaining part of this section, we will use a result of Haantjes (1) to prove the conformal invariance of J
IHI2dV
Rm.
for surfaces in
M
Assume that M
is a surface in
.
It is obvious
is invariant under similarity
(IHI2- G)dV
that the quantity
Rm
transformations. (i.e., motions and homothetics on
Rm.).
On the other hand, according to Haantjes [1], a conformal mapping on
Rm can be decomposed into a product of similarity Hence, it suffices to prove
transformations and inversions. that
Rm
an inversion on
on the surface of
is invariant under inversions.
(IHI2- G)dV
Rm.
such that the center of
We choose the center of
M.
Denote by
original surface respectively.
x
c
as the origin
i
M = #(M),
be the radius of inversion.
x = (°-2)x
r
r2 = <x,x>
,
From these we find 2
(3.25)
2
dx = (c2)dx - (2c ) (dr)x r
_ (3.26)
be
does not lie
$
2
(3.24)
$
the position vectors of the
x
and the inverse surface
M
Let
and
Let
r
(2 4
4)
Hence, the volume element 0 of
M satisfies
Then we have
§ 3. Conformal Invariants
211
4
dV = (cT) dV
(3.27)
r
Let
be orthonormal normal frame of
e3,...,e
M
Rm
in
Then
er =
(3.28)
2<x,e > (
r 2r
)x-e
are orthonormal normal frame of
r
r = 3,...,m From 13.25), (3.26) and
M.
(3.28), we find
(3.29)
r
2
2
- (cf)
ki(er) = -(!)ki(er) - (2 2 )<x,er>
(3.30) where
2c2<x,e > 4 r )
c
ki(er)
and Ae ,
and
are the principal curvaturs of
ki(er)
respectively.
Ae
From (3.30) we get
r
(3.31)
(kl(er)+ K2(er))`
4 k1(er)k2(er
(c4)((kl(er)+ k2(e r ))2 - 4 kl(er)k2(er))
By taking the sum of both sides of (3.31) over
IH12-G
_ =
(r44)(IHI2+_G)
.
c
Combining this with (3.27) we obtain
(3.32)
(IHI2- G)dV = (JH12- G2)dV
r,
we obtain
S. Total Mean Qwvature
212
This shows that
(IHI 2- G)dV
is invariant under conformal
mappings of
Rm
obtained in
G. Thomsen [1) in 1923.
.
R3
For surfaces in
(3.32) was already
,
M
If
is compact, (3.32)
and Gauss-Bonnet's formula gives the following. Proposition 3.5.
Let M be a compact surface in
a conformal mapping of
and
(3.33)
Remark 3.3.
J
Rm.
Rm
Then we have
IH12dV =
M
For surfaces in
R3,
Proposition 3.5
is due to Blaschke [1] and J.H. White [1).
The proof of
Proposition 3.5 above is a slight modification of Blaschke's.
§4. A Variational Problem Concerning Total Mean Curvature
213
A Variational Problem Concerning Total Mean Curvature
¢4.
In order to gain more information about the infimum of total mean curvature, one may apply standard techniques of We will deal with this variational
calculus of variations. problem in this section. Let
M
be a compact n-dimensional submanifold (with or
without boundary) of a Euclidean m-space Rm. the position vector of
M
Rm
in
where
ul,...,u
,
are local coordinates of
M
unit normal vector field of
in
Rm.
where
0
interval
(-c,c).
If
4
0
aM.
If
0
t
on the boundary
M
is a
.un)g(ul,...,un)
lies in a small
aM of
Rm
in
consider the normal variations which leave in the sense that both
S
We put
is a differentiable function and
(4.2) is called a normal variation of
on
If
M.
x(ul,...,un,t) = x(ul,...,un) + t4,(ul,
(4.2)
denote
Then
.
x = x(ul,...,un)
(4.1)
x
Let
aM
M,
We only
.
strongly fixed
and its gradient vanishes identically
M has no boundary, there is no restriction on
the normal variation.
Throughout this section, we put
and
xi = a2 i
gi7 = <xi,xi >.
Then the induced metric tensor on
by
g = E g dui®dui
.
M
is given
,
S. Total Mean Curvature
214
Let
denote the inverse matrix of
(g13)
dV
element
M
of
(4.3)
The volume
(gig).
is given by
dV = * 1 = W du1 n ... A dun
,
where
(4.4)
W =
If
det (gig
is a unit normal vector field which is in the
t
direction of the mean curvature vector (4.2) is called an H-variation of the operator
(a/at)Jt=p.
H-stationary if where
a
b
S
M
an dV = 0
6 f
M
Let
Rm
of
denote
b
is called
for all H-variations of
M,
M
denotes the mean curvature.
stationary if
Rm.
in
A submanifold
an dV = 0
then the variation
H,
And
M
is called
for all normal variations of
M.
M
It is clear that stationary submanifolds are H-stationary. If
M
is a hypersurface, an H-stationary hypersurface is always
stationary. Let
en+l " " 'em
vector fields on
M
be a local frame of orthonormal normal such that
en+l = g
define the natural orientation of
Rm.
and
xi,...,xn,en+l,...,em
Then we have
(_1W mfr
er =
(4.5)
where
[xi,...,xn,en+l'...,@r,.. ,em)
[v1,...,Vm-i) n
denote the vector product of
vl,...,vm-1
and
(4.6)
xij = va/aui(°a/auk x) = 2ax/auiau
denotes the omitted term.
Let
m -1
vectors
§ 4. A Variational Problem Concerning Total Mean Curvature
(4.7)
eri - a/aui(er) = aer/aui
215
.
Then the formulas of Gauss and Weingarten give
xij = E ijxk + E hijer
'4.8)
(4.9)
eri = -E hi
+ E Fries )
hr , =
where
i
s
(4.10)
and
D
9tJhij
=
Ls
is the normal connection.
From 14.2) we obtain by direct
computations that
6x = en+l
(4.11)
(4.12)
bxi = Oien+l -
(4.13)
bgi3 =
where
E hi+1 j xJ +
4i =
E Rn+l
i er
bgij = 24hn+lij bW = 4 (trace A
(4.14)
,
and
hrij
en+ 1 )W
git
htJ.
,
Moreover, we also
have
(4.15)
erij = -E
hik
xkj + E au
+ E Ari Rtj es
(is )es
(mod xk)
216
5. Total Mean Curvature
bxij o ijen+l + E
(4.16)
n+lk
-
+jtn+l i
s a hkj + + auj
r s n+1 i trjyes
+
where
hi
j+
erij = 62er/auiauj
(s tn+1 i) (mod xk)
and ij = a2Vauiauj.
,
Hence we
have
hi+l
(4.17)
<en+1'bxij> = 'ij -
(4.18)
<er.6xi j> _ itn+l j + jtn+1 +
au
khkjl -
h+l k hkj
tn+l
(tn+l i ) +
tn+l i n+l 1
tsj ' r = n+2,...,m
From (4.12) we have
if r=n+1; (4.19)
<xi,ber> =
rn+l i'
if
r = n+2,. .,m
where we have used (4.5). From (4.5), we also have
(4.20)
<e8.b(W er)>
= (-1)m+r [xl....,xneen+I"..,@r,.. ,em.bes)
for
r ¢ s,
(4.21)
where we use the notation
[vie ...,vm) =
(-1)m-1
217
§ 4. A Variational Problem Concerning Total Mean Curvature for
m vectors
vl,...,vm
1m.
in
Since <er,ber> = 0,
(4.19) and (4.20) imply
(4.22)
ben+l
--
xi m-n
m m
+
(-1W
[x1,...,xn,en+2,...,em.berler
r=n+ 2
(4.23)
ber = -4 , '1n+1 xi A ,er,
(-1)m+r-1
+
(xl,..
W
sir
r = n+ 2,...,m
...em,besle. , gti
where
4t
and
Ln+1 = E
gti bn+r
l t.
From (4.8), (4.22) and (4.23) we find
(4.24)
<X., en+l
= - E 4)k rij m +
(-l)phij
E
W
s=n+1
(xl,...,xn,en+2,...,em,bes)
(4.25)
<xij,ber> _ -E 4) An+l krij (-l)m+r-lhsj
+
sir
W n
(xl,...,xn,en+l,...,er,...,em,bes)
Thus, by using (4.8), (4.16), (4.24) and (4.25), we find
218
5. Total Mean Curvature
hn+1 t hn+1 i tj
bhn+l = 0 ij is j
(4.26)
m
E
+
(-1)
m-n
fr
Rr
n+l i n+l j
s
s,
hi.
s=n+2
bhij
(4.27)
-i
r+1
+ 7 1n+1 i -
E hi+1 t hr
(fir
n+l i )
au .
+
r
r
k
is + E 1n+1i1's7 - 4)L Ln+Ik +
s
'r
i7
(-1)m+r-1 hs
ij
W
n 'xnen+l....,er,..
fxl'
,em.bes]
r = n+ 2....'m where i,j = i7 - E k Ilj
are the components of the Hessian,
k
of
vd4),
4
on
M.
From (4.13) and (4.26) we obtain
(4.28)
b(trace An+l)
-p4)+4)IIAn+1II2-02
m
E
(tr As)<en+l'Res>
s=n+2 where
Ar = Aer,
Similarly, we have
IIAn+1II2
= trace(An+1)
and
R2 = E en+1 thn 1'
§4. A Variational Problem Concerning Total Mean Curvature
219
b(tr Ar) = 4) tr(ArAn+1)+2Eg'3 4i tn+lj
(4.29)
+4Eg - E
g ]i
)i r
1n+l i; j+
s to+l
r
i tsj
(tr As)<er.bes>
s¢r
where
tsk rij'
(tsi)
tsi;j
From (4.28) and (4.29)
k we get
(4.30)
6(a2) = 2 ((tr An+1)(-&0+4) (IAn+iII2-4
12
n
m
E (tr Ar) [4'tr(Ar An+1) + 2
+
r=n+2 E g
+
sr
where Des
to+l i; j + 4)
E g1 3 to+l i tsj)
er
In particular, if the normal variation (4.2) is an H-variation, then
tr Ar = 0
for
r = n + 2,...,m.
Thus
(4.30) reduces to
ba =
(4.31)
12+4)
n(-l4-4)
IIAn+1II2)
Therefore, by applying (4.3). (4.14) and (4.31), we have
(4.32)
g
J
c dV = $ M
M
f-cn ac-l p4) +
n
,n Y
n
4) ac-l
12
11A n+lII2 - n4) ac+l)dV ,
c20
5. Total Mean Curvature
220
If we integrate by parts to get rid of the derivative of
0,
we find
(4.33)
$(Aac-l)dV
(ac-1 A4))dV = J
f
,
M
M
where the boundary terms one would expect after integration by parts vanishes because of our hypothesis on
4)
on
aM.
Combining
(4.32) and (4.33) we find
6
pM ac dV = pM { c Aac-1-
c-1 f2
n
J
nac+1+n ac-1 flAn+1II2)dV From this we see that
6 t ac dV = 0
for all
H-variations if
and only if
c
(4.34)
Aac-1
- cac-1 12 - n2 ac+l + c
In particular, if
c = n,
Rm
only if the mean curvature
Aan-1
(4.35)
where
An+l
Then M
.
a
IIAn+1II2
=0.
this gives us the following.
Theorem 4.1 (Chen and Houh [1)). compact submanifold of
ac-1
M be an n-dimensional
Let
is H-stationary if and
satisfies
- an-1 { l2 + na2 - IIAn+
112) 1
=0
,
denotes the Weingarten map with respect to the unit
vector in the direction of
H.
From Theorem 4.1 we obtain immediately the following.
221
§ 4. A Variational Problem Concerning Total Mean Curvature
If an n-dimensional compact submanifold
Corollary 4.1.
M
Rm
in
is stationary, then
Aan-1
- an-1(.t2 + na2 - IIAn+1112 ) = 0 (Chen (3))
Corollary 4.2.
compact hypersurface in
3R n+1.
.
M be an n-dimensional
Let
Then M
is stationary if
and only if
(4.36)
pan-l+n(n-1)(an+l-an-1
where
T
T) = 0
denotes the scalar curvature of
Remark 4.1.
M
If
is a surface in
M.
R3
Corollary 4.2
,
is already known to Thomsen [1).
Using Theorem 4.1, we may obtain the following. (Chen and Houh [1)).
Theorem 4.2.
Rm.
(or H-stationary) submanifold of
mean curvature vector if and only if
Rm
submanifold of
of
Let
M be a stationary
Then M has parallel M
is either a minimal
or a minimal submanifold of a hypersphere
Rm
M
If
Proof.
is stationary or H-stational, then by
Theorem 4.1 we have (4.35).
If
M has parallel mean curvature,
then 0 = DXH = DX(a en+1) = (Xa)en+1+aDXen+1. constant.
Rm.
If
if
a = 0,
a ¢ 0,
then implies
then
then
M
Thus,
a is
is a minimal submanifold of
Den+1 : O.
Thus
l2 = O.
Equation (4.35)
5. Total Mean Curvature
222
0=
= n E (ki
no2
k]) 2
i< ]
where
are the eigenvalues of
kl,...,kn
is pseudo-umbilical in
M
that
Rm.
Proposition 4.4.2, we conclude that hypersphere of
Rm.
Theorem 4.3.
An+l.
This shows
Therefore, by applying M
is minimal in a
The converse of this is trivial.
(Chen and Houh [1]).
(Q.E.D.)
The only compact
pseudo-umbilical stationary (or H-stationary) submanifolds (without boundary) of
hypersphere of Proof. or
Rm
Since
no2 = IIAn+1112.
Rm
are minimal submanifolds of a
.
M
is pseudo-umbilical, we have either
Thus, Theorem 4.1 implies
pan-1
(4.38)
-Ao2n-2
=
- an-1 2 2 = O
2a2n-2- 2+
.
Thus (4.37) implies
the mean curvature vector Proposition 4.4.2,
of
Rm
M
a
k2 = 0.
is parallel.
H
0
Ildan-lilt
Thus, by Divergence Theorem, we conclude that which is non-zero.
is a constant
And hence
consequently, by
is a minimal submanifold of a hypersphere
(Q.E.D. )
.
Theorem 4.4.
a = 0
(Chen [3]).
stationary hypersurfaces in
The
Rn+l
only are h
odd-dimensional com act ers heres.
223
§4. A Variational Problem Concerning Total Mean Curvature Proof.
Let
be an odd-dimensional compact stationary
M
hypersurface in
Then Corollary 4.2 implies
Rn+l.
(4.39)
pan-l+n(n-1)(an+l-an-l r)=0
be the principal curvatures of M
kl,...,kn
Let
in
Rn+l
Then
a
n1
tan-1
n(n-1)an-1
tan-l
2
(T -a 1 < 0
= O.
by Proposition 4.4.1,
M
Rn+1.
is a hypersphere of
Theorem 4.5. (Chen (3).)
n
If
compact stationary hypersurfaces in
.
This implies that
is totally umbilical in
M
.
Thus
=
By Hopf's lemma, we obtain i.e.,
2 T= n n-1) kk i<7 i
a2 -, 2 O.
Therefore, we have
T = a2,
!
(kl + ... + kn )
Consequently,
Rn+1
(Q.E.D.)
is even, then the only
Rn+l
such that the mean
curvature does not change sign are hyperspheres. This theorem can be proved in a similar way as Theorem 4.4. Remark 4.2.
Theorem 4.1 shows that minimal submanifolds
of a hypersphere of
Rm
.
IRm
are H-stationary submanifolds in
However, there are many other H-stationary or stationary
compact submanifolds which are not of this type.
consider the anchor ring in
R3
given by
For example,
5. Total Mean Gbrvature
224
x1 = fa+bcosu)cosv, x2 = (a+bcosu)sinv, x3 = bsinu By direct computation, we have
a+ 2b cos u
a = 2b (a+b cos u) '
G=
cos u br
r = a + b cos u. Then we have
where
a2 r+bcosu
2a(a2-G) =
4b3r Moreover, we also have
&a =
-atb+acosu) 2b2r3
Consequently, we find
pa+ 2a(a2 - G) = a(a 2 - 2b2 ) 4b3r3
(4.40)
This shows that the anchor ring is stationary if and only if a -
b.
Remark 4.3.
M be a surface in a 3-dimensional
Let
Riemannian manifold
(M,g).
M with the induced metric change of metric on
M
obtained from
g*.
M.
Denote by g.
If
Denote by
g*
P-2
a
.
is a conformal
the induced metric on
Then the Laplacian
A* =
the Laplacian on
q* = p 2-g
satisfies
(4.41)
A
p*
of
(M,g*)
225
§4. A Variational Problem Concerning Total Mean Ckirvature
Define an operator
= A+(2a2-11h,12)I
(4.42)
M
Then
M by
on
M
is stationary in
.
if and only if a = 0.
From
(3.11) and (4.42), we find
(4.43) *
where
M with respect
denotes the corresponding operator on
*
to
g
Remark 4.4.
Weiner [1) defined a surface M
3-dimensional Riemanifold manifold (IHI2 5
J
integral
+ R')dV = 0
3.4), the equation
Aa2 + a (2a2 - ItA3112) = 0
the stereographic projection from
M + R3
f :M 4 S3
Weiner (11.)
Since the
M.
is itself invariant
In particular, if S3
onto
R3,
a
denotes
then an a
of
Thus, by using Lawson's examples of
compact minimal surfaces in R 3
in
is stationary if and only if
is stationary.
surfaces in
M
is a conformal invariant, (Proposition
under conformal transformations.
immersion
to be stationary if
for any variation of
(JH12+ R')dV
J
M
in a
S3,
we obtain compact stationary
of arbitrary genus.
(cf. Corollary 1 of
In fact, the stationary anchor ring given in
Remark 4.2 is one of the examples given by the sterographic projection in which the minimal surface in
S3
is the square
torus (or called the Clifford torus.) Remark 4.5.
For further results in this direction, see
J. Weiner (1), Willmore and Jhaveri [1).
5. Total Mean 04rvature
226
Rm
5. Surfaces in
which are Conformally Equivalent to
a Flat Surface The main purpose of this section is to improve inequality (2.1) on total mean curvature for certain compact surfaces in
an arbitrary Euclidean m-space Rm.
According to Nash's
Theorem, every compact Riemannian surface can be isometrically
Rm
imbedded in a
for large
Definition 5.1.
m.
M
A compact surface
in
Rm
is called
conformally equivalent to a flat surface if it is the image of a compact flat surface under a conformal map of
is equivalent to a compact flat surface up to
M
i.e.,
Rm,
conformal maps or diffeomorphisms of
Rm .
For such surfaces we have the following best possible result.
(Chen [9,191)
Theorem 5.1. in
Rm
Let
M be a compact surface
which is conformally equivalent to a flat surface.
Then we have 1H12dV
(5.1)
f
> 2n2
.
The ecuality sign of (5.1) holds if and only if conformal Clifford torus, i.e.,
M
M
is a
is conformally equivalent
to a square torus. Proof.
surface in
Since the total mean curvature of a compact
Rm
is a conformal invariant (Propositions 3.4
and 3.5), it suffices to prove the theorem only for compact
§5. Surfaces in mm which are Conformally Equivalent to a Flat Surface
flat surfaces in by A
in
p
M,
we denote
A.
Then we
the map;
(5.2)
A
by A(e) = Ae. have
For each point
Itm.
227
-4 End(Tp(M),Tp(M))
Tp`(M)
:
Let
denote the kernel of
Op
dim Op 2 m-5.
Denote by
the subspace of
Np
Tp(M)
given by
Tp(M) = N p O 0 p ,
Then we have
for any
A(e) = 0
orthonormal normal frame e 6 ,...,e m E 0 p
N p 1 Op
in
e
e3,...,em
at
Op.
p
We choose an in such a way that
Then, for each unit normal vector
.
e
at
t.,,
m
e = E cos 9 e r
(5.3)
r=3
Thus the Lipschitz-Killing curvature at
(5.4)
G(p,e)
(p,e)
is given by
E cos erh11)( E cos 6sh22) -( E cos eth12)2 r=3
s=3
t=3
The right-hand side of (5.4) is a quadratic form on Hence, by choosing suitable
(5.5)
e3, e4, e5
at
p,
we have
G(p,e) = a1(p) cos2e3+x2(p) cos264 + X3(p) cos2e5
al 2 a2 2 a3 Moreover, since
(5.6)
cos er.
M
is a flat surface, we find
)'1+a2+a3 = 0 1
aA = det(A2+A)
.
5. Total Mean CLrvature
228
In particular, we have cases
0
)`2
Case 1:
(5.7)
and
and
)`1 2 0
We consider the
separately.
X2 < 0,
alP l2 2 0.
)'3 S 0.
From (5.6) we have
G(p,e) = 11(cos2e3 -cos2e5) +'X2(cos204
-cos2e5)
.
Hence, (5.8)
!
IG(p,e)Jda =
sp J
111(cos2e3 S
- cos2e5)
+
))2(cos204
- cos2e5) Ido
P
X1(p)
i cos2e3 J
S
cos285Ida
P i cos2e4 - cos2e5Ida
+ X2(P) J
.
Sp On the other hand, by a formula on spherical integration, we have
(5.9)
fS p
I
Cos20r
- Cos2es 1da
= 2cm_1/tr2 ,
Hence, by (5.8) and (5.9) we find 2c
(5.10)
J
IG(p.e)lda
By the definition of
H,
m- 1
we have
(Xl(p)+ )L2(P))
r -/ s
§ 5. Surfaces in IRm which are Conforrnally Equivalent to a Flat Surface
(5.11)
4IHI2
=
h3 +h22)2+ (hll+h22)2+ 11
229
(hll+h22)2
(311)2+ (h22)2+2x1+2(hi2)2
4(x1+)6 2) combining (5.10) and (5.11), we obtain 2
(5.12)
where
IHI2
G*(p) =
Case 2:
J
(p)
2cm_1
IG(p,e)Ida. S
p
From (5.6) we find
x2, x3 < 0.
G(p,e) = x2(cos204 -cos2g3) +x3(cos295 -cos2g3) Thus, from (5.6) and (5.9). we obtain
(5.13)
G*(p) S -x2
J
,Icos294 - cos203Ida Sp
Icos2a5 - cos293Ida
- X3 J s
p
2 x1cm-1/rr2 On the other hand, we also have
4IHI2
(hi1)2+ (h2?)2+ 2X1+ 2(hi2)2 2 4x1+4(h12)2
4x1
.
S. Total Mean Curvature
230
Hence, we get
2cTF
(5.14)
G* (p)
JHJ2 2
M-1
Consequently, we obtain (5.12) in general.
Thus, by taking
integration of both sides of (5.12), we obtain 2
(5.15)
fM
IHI2dV _
by virtue of Theorem 4.7.1.
M
i b(M) M
Now, since
is compact and flat,
is either diffeomorphic to a 2-torus or diffeomorphic to b(M) = 4.
a Klein bottle. In both cases, we have
Thus (5.15)
implies (5.16)
IH
S
12dV
, 2,2
.
If the equality of (5.16) holds, then the inequalities in (5.8) and (5.13) become equalities. of of
1
2
and and
X
2
X3
Hence, at least one
is zero for the first case and at least one is zero for the second case.
implies that the second case cannot occur. X2 = 0
identically on
However, this
Thus, we find
Furthermore, since the equality
M.
signs of (5.11) hold, we have 3 3 h11 = h22 ,
41
Now, because
= 0, )`2
=
42
4 3 h12 = h12 = 0
51 ,
these imply
+
52
= O
.
§5. Surfaces in IRm which are Con formally Equivalent to a Fiat Surface
h3
(5.17)
11
h3
= h3 22' ,
12
= 0,
h4
i.
231
h5 + h522 = 0
= O,
11
Consequently, by choosing suitable orthonormal frame we have
el,e2,e3, ..,em,
(5.8)
A
=
3
a
O
0
a
a A
4
=
0
In particular, by Proposition 4.3.2,
Rm.
connection in
,
-a0
A
5
= ... = A m = 0 .
has flat normal
M
Thus, by applying Proposition 4.3.1, we
see that, there exist locally orthonormal normal frame e3,...,em
such that
= Dem = 0
De3 =
(5.18)
.
We put (5.19)
er = E arses ,
is an orthonormal
(ars )
Then
Since
(x,y)
M
a/ax
and
Denote by 6/by,
L = h(X1,X1) ,
(5.20)
and
is covered by an isothermal coordinate
vX X _ i
M has the form
such that the metric on
g = E(dx2+ dy2). fields
(m - 2) x (m - 2) - matrix.
is two-dimensional and our study is local, we
M
may assume that system
r = 3,...,m
:i Xk.
X1
and
X2
respectively.
M = h(X1,X2) ,
Then we have
the coordinate vector
We put
N = h(X2,X2)
5. Total Mean Curvature
232
= X1E/2E r111 = r12 = -r22
(5.21)
Therefore, the Codazzi equation reduces to
L - DX M = (X2E )H ,
DX
1
2
(5.22)
DX M -D N = -(X,E)H . x1 2 Since
X1
and X2
are orthonormal, we may define a function
by
e = e(X,Y)
X1 = cos 9 e1+sin 9 e2 , (5.23)
X2 = -sine el + cos g e2 With respect to the frame field
.
X1, X2, e3,...,em,
the second
fundamental tensors are given by
a(arl +a r2 cos 20) A
Since M
r
-aar2 sin 2e
`
r=3....,m,
=
-aar2 sin 29
is flat, we have
E = 1.
a(arl - ar2 cos 29) Thus, by (5.18), equation
(5.22) of Codazzi reduces to
(5.24) (5.25)
ay (a(arl+ ar2 cos 2e)]
a (aar2 sin 2e1
-ay (aar2 sin 2 e] = a [a(arl- ar2 cos 29))
§5. Surfaces in IRm which are Confornwily Equivalent to a Flat Surface
Multiplying
arl
to (5.24) and summing over
using the fact that
(5.26)
a
In a
=E(
(ars) E O(m -2),
as
r,
233
then, by
we get
as
ayl)ar2 cos 29+( axl)ar2 sin 29)
Similarly, multiplying
arl
to (5.25) and summing over
r,
we have
(5.27)
I In a = E ( (asay)ar2 sin 29 - (asa_Jr')ar2
Multiplying
ar2
to (5.24) and summing over
cos 29) )
r,
we find
ar2(aayl) + (a axn a)sin 29 + (a ay a )cos 29
2 ay sin 29 - 2 ax cos 29 By substituting (5.26) and (5.27) into this equation, we get
(5.28)
L ar2 aayl = sin 20
Similarly, by multiplying
ar2
" - cos 28 ax
to (5.25) and summing over
r
and using (5.26) and (5.27), we get
(5.29)
E ar2 aaxl = -cos 29y - sin 29ax
Substituting (5.28) and (5.29) into (5.26) and (5.27), we may
find Ana = 2&
ax
by '
a Ana =-be by
ax
S. Total Mean Curvature
234
From this we get 2
2
( + a2)(En a) = 0
(5.30)
6x
(5.30) implies
E = 1,
Since
A in JHI2 = A in a2 = 0 .
(5.31)
Because and
M
by
is a non-negative differentiable function on
IHJ 2
is compact, (5.31) implies that
is a positive
We put
constant (cf. Yau [1)).
e
IHJ
M
= cos g e3+ sin g e4
(5.32)
e5 = e5""' em = em
e4 = sin 9 e3 - cos 8 e4 , el,e2,e3, ..,em,
With respect to
(5.33)
A3 =
C
O
O
a
we have
O
A5 = ... = Am = 0
A4 =
O
O
O
2a
From (5.33) and the structure equations of Cartan, we may easily find that both the distributions are parallel.
see that
Thus, by the deRham decomposition theorem, we
M = C1 xC2,
manifold of
Ti.
where
C.
is the maximal integral
Moreover, because
of Moore (1] implies that C1
T. = []R ei), i = 1,2,
is in a linear r-space
m (m - r) - space R-r
.
h(el,e2) = 0,
a result
is a product submanifold where
M IR
r
and
C2
is in a linear
Thus, by a result of Kuiper
[ 1) ,
the
§5. Surfaces in IRm which are Conformally Equivalent to a Flat Surface total absolute curvature of
M
and
total absolute curvatures of
C1
absolute curvature of
Rm
M
in
C2,
2.
is the product of Because the total
C2.
is equal to
the total absolute curvatures of to
Rm
in
and
C1
by (5.14),
4
are both equal
C2
Hence, by applying Fenchel-Bosuk's result to
we conclude that both
and
C1
are circles of the same radius.
R4
torus in a linear 4-space
.
and
C1
are planar curves with
C2
curvature .a which are constants.
235
Therefore,
Consequently,
C1
M
and
C2
is a square
The converse of this is clear. (Q.E.D.)
Remark 5.1. M
by
is a flat torus in
M
If
is homothetic to the flat torus (1,0)
and
with
(x,y)
0 < x
RX /I', 2
and
Rm with y 2
such that r
generated
J1- x2
,
Li
and Yau (1) obtain very recently the following inequality.
(5.34) Remark 5.2.
f
IH12dV
',2(y+y)
M
From Theorem 5.1, we obtained immediately the
following.
Corollary 5.1.
M
If
is a compact surface in
Rm
is conformally equivalent to a Klein bottle, then we have
JHI2dV >
(5.35)
M
27r2
which
5. Total Mean Curvature
236
Surfaces in
*6.
R4
The main purpose of this section is to improve inequality
R4
(2.1) on total mean curvature for certain surfaces in
Rm.
Let M be a compact surface in orthonormal normal frame
M in
of
be a normal vector of M
cos grey
e =
e3,...,em
Ae = E cos grAr.
We choose an
at
Rm.
Then we have
p.
Thus
G(p,e) = det(Ae) = det(E cos grAr)
(6.1)
Let
.
Since the right-hand side of (6.1) is a quadratic form of cos 03,...,cos gm.
Thus we may choose a suitable local orthonormal
e3,...,em
normal frame
such that with respect to
we have
er,
m 2 E "r_2 cos9r ,
(6.2)
r=3
Al -' 12
We call
We call such a frame as Otsuki frame.
the A-th curvature of the surface M surface in ll
and by
R4
,
p
the second curvature
Theorem 6.1.
R4
If
we simply denote by
(Chen [9))
H V( 0,
lA, A = 1,2,...,m-2, If
M
is a
the first curvature
).
X a.
Let M be a compact surface in
M has non-positive Gauss curvature, then we have
(6.3)
If
Rm.
in
... 'm-2
P
JH M
j2dV
k 2,r2
.
then the equality sign holds if and only if
a square torus in
R4
.
M
is
§ 6. Swjaces in IR
R4
M be a compact surface in
Let
Proof.
237
BI
the bundle of all unit normal vectors of M
by
Tr:B1 4 M
the projection of
onto
BI
Denote by
.
in
R4
and
Let
M.
Then by the hypothesis, we have
W = (p E M IX(p) Z 0).
JG(p,e)J = 1% cos2e+ p sin2el
(6.4)
= IX cos 2e+ G sin2el S XIcos 291 -G sin2a
on
where
it 1fW),
an Otsuki frame of
(6.5) On
J
-l ir
and
e = cos a e3 + sin a e4
M
in
R4
.
e4
form
Thus, we find
IG(p,e) JdV Ado S 4 (W)
e3
J
)LdV -ir W
J
GdV W
Bl - rl(W), we have
(6.6)
J
Bl-tr -1 (W)
IG(p,e)IdV Ado = -f
Bl-?r
= -r f
1
G(p,e)dV Ado (W)
G(p)dV . M-W
On the other hand, by definitions, we have on
(6.7)
41HI2
= E (tr A r=3
r
)2 = 2G+ E IIA
r=3
2 2(a+µ)+2(). -4) = 4k Thus, we get
.
r
W,
5. Total Mean Curvature
238
(6.8)
IHI2dV , f
f M
dV W
4 f _l n
+
IG(p,e)IdVnda+4 f 4
(W)
4
fBl-r-1(W)
+
G dV W
IG(p,e)IdV n do
4 f M-W
G dV
= 4 f B IG(p,e)IdVAdo+4 f M
G dV
2
2 2 (b(M) + X(M)) = 2r2 This proves inequality (6.3).
If the equality sign of (6.3)
holds, then all the inequalities in (6.4) - (6.8) become equalities. W = M. in
R4
Assume that
then (6.8) implies that
IHI > 0,
Moreover, from (6.7), we see that .
M
is pseudo-umbilical
Furthermore, from (6.4) we find
I%cos20+Gsin26l = alcos 2aI -G sin2a
(6.9) for all
q.
Thus
G = 0,
i.e.,
M
is flat.
from the proof of Theorem 5.1, we conclude that
torus in
R4
Consequently,
M
is a square
(Q.E.D.)
.
Combining Corollary 3.1 or Proposition 3.5 with Theorem 6.1,
we obtain immediately the following. Corollary 6.1.
Let
M
be a compact surface in
Rm
which is conformally equivalent to a compact surface in
R4
§6. Surfaces in !R' Then we have
with non-positive Gauss curvature.
(6.10)
239
JH12dV 2 2r2
1
.
M
From Chen (9], we also have the following. Theorem 6.2.
M
Let
R4
is conformally equivalent to a compact surface in non-negative Gauss curvature.
(6.11) then
M
which with
If we have
JH12dV s (2+Tr)rr
S
Rm
be a compact surface in
,
is homeomorphic to a 2-sphere.
For the proof of this theorem, see Chen (9]. Let
f : M - R4
surface into necessary,
R4
be an immersion of an oriented compact By applying regular deformation to
.
intersects itself transversally, thus,
f(M)
At each point
intersects itself at isolated points. self-intersection, we assign
+1
if
f
p
f(M)
of
if the direct sum orientation
of the two complementary tangent planes equals to the given orientation on
R4
,
and we assign
-1
otherwise.
Then the
self-intersection number is defined as the sum of the local contributions from all the points of self-intersections.
known that the self-intersection number invariant up to regular homotopy of the following result of Theorem 6.3.
M
If
It is
is an immersion
into
R4
.
We mention
S. Smale (1] for later use.
Two immersions of
S2
into
R4
are
regularly homotopic if and only if they have the same self-
5. Total Mean Q rvature
240
intersection number. This theorem says that the self-intersection number is the only regular homotopic invariant of
R4
For surfaces in Theorem 6.4
denotes the genus of
Proof. in
M
R4 and
f : M .. R4
R4
into
M.
such that, restricted to are normal to
and the normal curvature
GD
be an immersion
,
We choose an orthonormal local frame
e3,e4
R4
Then we have
.
4ir(l+ IIf I -g)
IHI2dV SM
g
Let
M
of a compact oriented surface
where
in
we also have the following.
,
(Wintgen (2]).
(6.12)
S2
el,e2
M,
el,e2'e3'e4
are tangent to
Then the Gauss curvature
M.
are given respectively by
G - R(el,e2;e2,e1) = E (h11h22 -(h12)2) GD = RD(el,e2;e4,e3) - h12(h22 - hll)
-h 412(h22
- hll
Thus, the mean curvature vector satisfies
IHI2
1 ((h131+h22)2+ (h411 +h22)2) 2
h3
112 h3 22
+
h4 11
h4
22
2
2
E 4
3
3
3
4
Ih1 21 Ih11 - h221 + Ih12I 1hll IGDI +G
.
+
(h12)2+ (h12)2+G 4
- h221 + G
G
§ 6. Surfaces in !R
241
Hence, we have
(6.13)
J
IHI2dV > f
G dV
IGDIdV + f
.
M
M
M
It is known that the integral of the Gauss curvature G and the integral of the normal curvature
2rr X(M)
2rr XD(M),
where
f
gives
denotes the Euler number of the normal
XD(M)
bundle (see, for instance, Little (1]).
(6.14)
GD
gives
Thus, (6.13) implies
IHI2dV > 2?r()((M) + IXD(M) I) M
On the other hand, by a result of Lashof and Smale (1), we have
XD(M) = 2 If.
Thus, by (6.14), we obtain (6.12).
(Q.E.D.)
Combining Theorems 6.3 and 6.4, we have the following.
Theorem 6.5. (Wintgen' [2]) . Let f -S 2 -. R4 immersion of a 2-sphere S2 into R4 . If (6.15) then S2
f
be an
IHI2dV < 8tr ,
is regularly homotopic to the standard imbedding of
f
into a linear 3-space
R3
If f :M -+ R4 M
R4
into
R4
- f(M)
,
is an imbedding of a compact surface the fundamental group irl(R4 - f(M)) of
is called the knot group of
of generators of knot group of of
f.
f
f.
The minimal number
is called the knot number
Wintgen obtained the following relation between total
mean curvature and knot number:
242
5. Total Mean Curvature Theorem 6.6 (Wintgen (1)).
Let
imbedding of a compact surface
M
R4
into
IH 12dV
(6.16)
be an
f : M -* R4
Then we have
.
47r p
M
where
p
denotes the knot number of
Proof.
We need the following simple lemma:
Lemma 6.1.
R4
f.
Let
be a height function of
ha
m
in
which has only non-degenerate critical points on
Then the number
(30(k)
of local minima satisfies
00/ha) -> p.
Without loss of generality we can assume that different values at the critical points written in the order induced from
ha.
M.
ha
takes
pi (i = 0,1,...,t) Let
c.
be real
numbers with
c0 < ha(p0) < c1 < ha(pl) < ... < ha(pt) < at+1
By a result of van Kampen for the fundamental groups of the spaces
Hi = (p E R4 - M I
we have
'rl(Hj+l) p ,r1(Hj) + one generator, if
7T 1(Hj+1)
pj
is a local minimum;
,rI(H.) + one relation, if
irl(Hj+l) N Tr1(H.),
if
pj
pj
is a saddle point;
is a local maximum.
243
§6. Surfaces in JR" The lemma follows from these relations.
We denote by Since
A2(ha) =
A2(ha)
Moreover,
Ae
in
B1
p2(ha) Z p.
For
ha,
of
a
is semi-definite if
or local minimum at (p,e)
p
ha.
Lemma 6.1 implies
13 0(h-a),
each critical point
the number of local maxima of
p.
such that
Let Al
U
is normal to ha
U
at least
2p
at
p.
is either local maximum
denote the set of all elements
is semi-definite.
to above observation, we see that the unit sphere covered by
M
times under the map
Then according S3
is
v : B1 ..
S3
Thus, by a similar argument as given in the proof of Theorem 2.1, we obtain (6.16).
Remark 6.1.
(Q.E.D.)
For a surface in
Theorem 1.2 if knot number is
2 3
I23,
Theorem 6.6 improves
and, for a surface in
Theorem 6.6 improves Theorem 2.1 if the knot number is
it4
2 2.
Remark 6.2.
Lemma 6.1 is essentially due to Sunday [1).
Remark 6.3.
Theorems 1.2, 6.2, 6.4, 6.5 and 6.6 can be
regarded as partial solutions to Problem 2.2.
5. Total Mean Curvature
244
Surfaces in Real-Space-Forms
*7.
Let
f :M -. FP(c)
M
oriented surface c.
be an isometric immersion of a compact
into a real-space-form of constant curvature
By Ricci's equation, the normal curvature tensor
RD
satisfies
RD(X,Y)g = h(X,A9Y)-h(A9X,Y)
(7.1)
X, Y
for
tangent to M and
g
We put
be an orthonormal tangent frame. i , j = 1 , 2 .
We define
normal to
Let
(X1,X2)
hij = h(Xi,X
as the endomorphism
a A b
(aAb)(c) =
(7.2)
M.
.
Then (7.1) becomes
RD(Xl,X2) = (h11-h22) Ah 12
(7.3)
The mean curvature vector
.
and the Gauss curvature
H
G
given by
(7.4)
4IHI2 = Ih11 + h22I2
For each point p in (7.5)
If
Ih12I2
G =
,
M.
We put
Ep = (h(X,X) IX E Tp(M), IXI = 1)
X = cos 0 X1+ cos 8 X2,
then
+c
are
245
§ 7. Surfaces in Real- Space- Forms
h(X,X) = H+ cos 2e h11-h22
is an ellipse in the normal space
This shows that
Ep
centered at
Moreover, as
H.
Tp(M)
goes once around the unit
X
We
goes twice around the ellipse.
h(X,X)
tangent circle,
+ sin 2e h12
notice that this ellipse could degenerate into a line segment on a point. at
p.
at
p.
we call this ellipse
The ellipse
RD # 0,
If
is degenerate if and only if
Ep
then
the ellipse of curvature
EP
h11-h22
and
h12
and we can define a 2-plane subbundle
are linearly independent
N
of the normal bundle
This plane bundle inherits a Riemannian connection
T.L(M).
Let
be an orthonormal oriented
from that of
T1'(M).
frame of
We define the normal curvature
TP(c)
N.
(e3.e4)
GD
of
M
in
by
(7.6)
G- =
Since M N'
RD = O
and
N
are oriented,
GD
is globally defined.
be the orthogonal complementary subbundle of
N
in
Let T1(N).
Then we have the following splitting of the normal bundle; TA. (N) - N ® Nl.
(7.7)
From the definition of
RD(X1.X2)S = 0
Let
a0 = a0(M)
of the tangent bundle
if
we have
N1.
P, E N1
.
denote the bundle of symmetric endomorphism T(M).
Define a map
4 :N -. a0 by
5. Total Mean Curvature
246
tr A A
(7.8)
Thus, because
RD
0
-
$ E N
2F
by assumption,
A
[A
e3.
(7.8) implies that
4
e
.
] ¢ 0.
Thus
We denote by
is an isomorphism.
X(N)
N
the Euler characteristic of the oriented 2-plane bundle over
We mention the following extension of a result of
M.
Little [1], Asperti [1] and Dajczer [i); Proposition 7.1
(Asperti-Ferus-Rodriguez [1]).
For a
compact, oriented Riemannian surface M isometrically immersed in a real-space-form
with nowhere vanishing normal
Mm(c)
curvature tensor, we have
X(N) = 2%(M)
(7.9)
be the bundle of symmetric
a0 = a0(M)
Let
Proof.
.
endomorphism endowed with the orientation induced by that of
N
via
Then because
4.
isomorphism, we have B(X)
let
is an orientation-preserving
4'
For each
X(N) = )((a0(M)).
be the element in
at
a0(M)
p
given by
B(X)(Y) = 2<X.Y>X - <X,X>Y
.
B(cos tX + sin tXl) = cos 2t B(X)+sin 2t B(X)l,
Then
Xl
X E Tp(M),
is a vector in
X, X1
Tp(M)
such that
give the orientation of
M.
IXjI = IXI,
where
X 1X1
and
Therefore, the index
formula for the Euler characteristic applied to a generic vector field proposition.
X
and to
B(X),
respectively, yields the (Q.E.D.)
§ 7. Surfaces in Real- Space- Forms
247
The following result is a generalization of Theorem 6.4. (Guadalupe and Rodriquez [11).
Theorem 7.1.
be an isometric immersion of a compact oriented
f :M -o Mmfc)
surface Mm(c).
Let
into an orientable m-dimensional real-space-form
M
Then we have
f
(7.10)
IH12dV
2 27 X(M) +
GD dVI -c vol(M)
If
.
M
M
GD
The equality holds if and only if
does not change sign
and the ellipse of curvature is a circle at every point.
From (7.1) and (7.6) we have
Proof.
D =
(7.11)
G
Ih11 - h22IIh12I
Thus, (7.4) and (7.11) imply
( Ih11 - h221 - 21h121 )2
0
Ihll =
-h2212+41h1212
-41h11 -h221 Ih121
Ih1112+ 1h22I2+21h1212-2G-4IGDI+2c IlhIl2-2G-4IGDI+2C
.
On the other hand, 41HI2
Ih11I2 + (h22I2+ 2
Ih11+ h22I2 =
1 h 1 1 1 2 + (h221
= 1Ih1I 2 + 2G
- 2c
2+ 21h1212+ 2G- 2c .
5. Total Mean CLrvature
248
Hence, we find
IHI2+ c _> G+ IGDI
(7.12)
with equality holding if and only if
2 (h11 - h22) = h12'
i.e., the ellipse of curvature is a circle.
M
over
gives (7.10).
if and only if
Integrating (7.12)
Moreover, the equality of (7.10) holds
does not change sign and the ellipse is
GD
always a circle.
Corollary 7.1.
be a compact oriented surface immersed in GD > 0
curvature
Let
(Guadalupe and Rodriguez [1)).
R4
M
If the normal
.
everywhere, then
(7.13)
IHI2dV
12ir
.
The equality holds if and only if the ellipse of curvature is always a circle. Proof.
Thus
M
If
GD > 0
X(N) = 2n
everywhere,
is homeomorphic to
Hence,
S2.
J
GD dV > 0.
X(N) = 2X(M) = 4,
which yields (7.13) by using (7.10).
Remark 7.1.
(Q.E.D.)
Atiyah and Lawson (1) have shown that an
immersed surface in
S4
has the ellipse always a circle if and
only if the canonical lift of the immersion map into the bundle of almost complex structure of
S4
is holomorphic.
curves in this bundle can also be projected down to to obtain examples of surfaces in
S4
Holomorphic S4
in order
with the property that the
ellipse is always a circle, hence giving equality in (7.10).
SUBMANIFOLDS OF FINITE TYPE
Chapter 6.
Order of Submanifolds
§1.
It is well known that an algebraic manifold (or an algebraic variety) is defined by algebraic equations.
Thus,
one may define the notion of the degree of an algebraic manifold by its algebraic structure (which can also be defined by using homology).
The concept of degree is both important
and fundamental in algebraic geometry.
On the other hand, one
cannot talk about the degree of an arbitrary submanifold in IItm
.
In this section, we will use the induced Riemannian
structure on a submanifold defined numbers
Here
p
p
and
q
M
Rm
of
associated with the submanifold
is a positive integer and
integer
S p.
manifold
M
We call the pair (Chen [151,22,25]).
to be of finite type if
q
to introduce two well-
q
[p,q]
is either
+ .
or an
the order of the sub-
The submanifold M
is finite.
M.
is said
The notion of order
will be used to study submanifolds of finite type in sections 2 through 5.
It was used in sections 6 and 7 to study total mean
curvature and some related geometric inequalities.
The notion
of order will be also used to estimate the eigenvalues of the Laplacian of
M
in the last three sections.
The order of a submanifold is defined as follows.
M be a compact Riemannian manifold and
M acting on
C+(M).
Then
A
A
Let
the Laplacian of
is a self-adjoint elliptic
operator and it has an infinite, discrete sequence of eigenvalues (cf. 43.2):
6. Submanifolds of Finite Type
250
0 = )`0 < al < %2 ... < lk < ...
(1 .1)
t
Let Vk = (f E C '(M) I Of = lkf} be the eigenspace of with eigenvalue
Then
Xk.
Vk
We
is finite-dimensional.
define as before an inner product
(f,g) = f
(1.2)
a
(
,
)
on
C (M)
by
fg dV
M
Then E 0 Vk by 0 Vk
is dense in
COO(M)
the completion of E Vk,
(in
L2-sense).
Denote
we have (cf. Theorem 3.2.2)
C (M) ='kVk f E C(M),
For each function of
ft
Vt (t = 0,1,2,...).
onto the suspace
f
let
be the projection Then we have the
following spectral decomposition
f = E ft,
(1.4)
(in
L2-sense)
t=O
Because V0 function
there is a positive integer
f E C *(M),
such that
fp 1 0
and
tap
f0 E V0
is a constant.
which are nonzero, we put integer
(1.6)
p z 1
f - fO= E ft
(1.5) where
1-dimensional, for any non-constant
is
q,
q a p,
If there are infinite
ft's
q = . Otherwise, there is an
such that
fq V 0
q
f - fo = E ft t=p
and
§ 1. Order of Submanifolds
If we allow
q
to be
251
we have the decomposition (1.6)
W,
m
for any f E C (M). M
Riemannian manifold
(1.7)
IRm,
we put
x = (xl,...,xm)
where in
into
xA
1Rm
.
of a compact
x :M 4 IRm
For an isometric immersion
,
M
A-th Euclidean coordinate function of
is the
For each
we have
xA,
qA xA -(xA)
(1.8)
= O
tF
For each isometric immersion
(1.9) A
is easy to see that or an integer and
q
p
z p.
q = q(x) = sAup(gA)
A
ranges among all
such that
is an integer
it
xA - (xA) 1
and
q
It
71 O.
is either
Moreover, it is easy to see that
p
are independent of the choice of the Euclidean coor-
dinate system on
1Rm
.
Thus
p
and
q
are well-defined.
Consequently, for each compact submanifold M more precisely, for each isometric immersion have a pair [p,q]
.
we put
x : M + ]Rm ,
p = p(x) = iAnf(pA},
where
A = 1,...,m
At
(xA)
=PA
associated with
M.
the order of the submanifold
M.
[p,q)
in
]Rm
(or,
x : M + ]Rm), we
We call the pair
By using (1.7), (1.8) and (1.9) we have the following spectral decomposition of
x
in vector form:
252
6. Submanifolds of Finite Type
q
x = x0 + E xt
(1.10)
t=p
in Mm
A compact submanifold M
Definition 1.1.
said to be of finite type if
q
is
Otherwise M
is finite.
is
of infinite type (Chen (22,25])".
A compact submanifold M
Definition 1.2.
said to be of
k nonzero
k-type
(k = 1,2,3,...)
(t t 1)
xt's
For a submanifold M
that M
if there are exactly
of order
(or of order
called a submanifold of order
we sometime say
[p,q), s q)
A submanifold of order
is not considered.
is
in the decomposition (1.10).
z p
is of order
Rm
in
if
q
(or
is also
[p,q]
p.
Let M be a compact submanifold of Rm .
Remark 1.1.
k-type in Rm
It is easy to see that M
is of
of infinite type in
if and only if M is of k-type
in any
Rm+m DJRm
Rm+m
Rm)
M
Lemma 1.1.
3Rm)
Let x : M -0 ]Rm
Proof.
M
(resp.,
(reap., of infinite type in any
be an isometric immersion
of a compact Riemannian manifold M the centroid of
into
Rm.
Then
x0
Rm.
in
Consider the decomposition
x = E xt
(1.11)
t=O We have
p)
Axt = atxt.
If
t y/ 0.,
then Hopf lemma implies
is
253
§ 1. Order of Submanifolds
xt dv - -11
f
(1.12)
tM
M
Since
Ax t dV = O
is a constant vector in
x0
we obtain from
3tm,
(1.11) and (1.12) that x0 = f x dV / vol (M)
(1.13)
.
M
This shows that
is the centroid of
x0
M.
(Q.E.D.)
Lemma 1.1 shows that if we choose the centroid of (in
as the origin of
3tm)
3tm
M
then we have
,
q
(1.14)
x = E xt
t=p
Let
v1
and
v2
be two
Htm-valued functions on
We define the inner product of
(vl,v2) = f
(1.15)
and
vl
by
v2
< v1,v2 >dV
M.
,
M where
v1.v2.
We have the following.
Lemma 1.2.
Let
x :M -6 IRM
be an isometric immersion
of a compact Riemannian manifold M
(xt,xs) = 0
(1.16)
where
xt
is the
for
t-th component of
into
t ¢ s
3tm.
,
x with respect to the
spectral decomposition (1.10). Proof.
Since
A
Than we have
is self-adjoint, we have
6. Submanifolds of Finite Type
254
at(xt,xs) = (Axt,xs) = (xt,Axs) = Xs(xt,xs)
Because
at
i as,
we obtain (1.16).
(Q.E.D.)
§ 2. Submanifolds of Finite Type
255
Submanifolds of Finite Type
42.
First, we rephrase Proposition 4.5.1 of Takahashi in terms of order of submanifolds as follows: Proposition 2.1.
x :M -
Let
]m
be an isometric
M
immersion of a compact Riemannian manifold Then
x
is of
1-type if and only if
manifold of a hypersphere of
into
7Rm.
is a minimal sub-
M
]Rm
From this proposition, we see that if minimal submanifold of a hypersphere
M
SD-1(r)
is a compact centered at
the origin, then we have
(2.1)
for some constant
X X.
Because
Ax =
- nH
ap E 7R
.
(Lemma 4.5.1),
(2.1) implies
HH = X H,
(2.2)
In views of this, we give the following characterization of submanifolds of finite type (Chen (221). Theorem 2 . 1 .
Let
be an isometric immersion
x : M + 1 m
of a compact Riemannian manifold
M
into
]Rm.
Then
M
is
of finite type if and only if there is a non-trivial polynomial P
such that
(2.3)
P(6)H = 0
.
6. Submanifolds of Finite Type
256
In other words,
M
curvature vector
is of finite type if and only if the mean satisfies a differential equation of the
H
form:
AkH+c1Ak-1H+ ...+ck-lAH+ckH = O
(2.4)
for some integer
k ? 1
Let
Proof.
and some real numbers
x : M -0 IRm
c1....,ck.
be an isometric immersion of a
compact Riemannian manifold M
into
]Rm
.
Consider the
following decomposition
q x = x0 + E xt ,
(2.5)
Axt
t=p If
M
is of finite type, then
-nA1H =
(2.6)
t=p
q < ..
1 i+1 xt t
xtxt
From (2.5) we have
i - 0 , 1 , 2 ,...
,
q Let
E xt
cl
t=p (-1)q-p+l
lp
(2.7)
-
Xq.
c2
t<s ltls
cq-Pt 1
Then by direct computation, we find
AkH + c1 Ak 1H +
where k - q -p+ 1. k z 0,
'
... + ckH - 0 ,
Conversely, if H
then, because m
satisfies (2.7) for
is compact, we have k x 1.
the spectral decomposition (2.5).
Using (2.6) and (2.7), we
find
(2.8)
E It0t+c1xt'1+ ...+ek-llt+ek)xt - 0
t=1
Consider
§ 2. Submanifolds of FYnite Type
For each positive integer
t
at(Xk+clat-1+
1
Since
(x5.xt) =
J
<x s
(2.8) gives
s,
m
(2.9)
'x
t
257
... +ck) fM for
> dV = 0
0
t ¢ s
(Lemma 1.2),
we obtain (fig+c1ag-1+ ...+ck)11xs1j2
(2.10)
=0
where
Nall2 = (xs,xs)
(2.11) If
xs ¢ 0,
then
jjxsjj ¢ 0.
(2.12)
Thus (2.10) implies
+clas-1+
)
...+ck = 0 k
Since equation (2.12) has at most
real solutions and
equation (2.10) holds for any positive integer
k
of the
are nonzero.
xt's
s,
at most
Thus the decomposition (2.5) is
in fact a finite decomposition.
Consequently,
M
is of finite
type.
From the proof of Theorem 2.1, we also have the following. Let
Theorem 2.2. Then
M
is of
is a polynomial
k
k-type P
M
be a compact submanifold of Mm. if and only if there
(k =
of degree k
such that
distinct positive roots and
(2.13)
P(A)H = 0
.
P(t)
has exactly
6. Submanifolds of Finite Type
258
By using exactly the same proof as Theorems 2.1 and 2.2, we may also obtain the following. Let
Theorem 2.1'.
x : M -4 ]Rm
of a compact Riemannian manifold
M
be an isometric immersion into
Then M
IRm.
is
of finite type if and only if there is a non-trivial polynomial
such that
P(t)
P(A) (x -x0) = 0
(2.14)
Let
Theorem 2.2'.
M
Then
is a polynomial
k
be a compact submanifold of
M
k-type
is of
k
Rm.
if and only if there
(k = 1,2,3,...)
of degree
P
.
such that
P(t)
has exactly
distinct positive roots and
P(A) (x-x0) = 0
(2.15)
.
From the proof of Theorem 2.1, we see that
Remark 2.1.
the positive roots of
in Theorems 2.2 and 2.2'
P(t)
fact eigenvalues of the Laplacian of
are in
M.
The following corollary is an easy consequence of Theorem 2.1.
Corollary 2.1. If
then
be a compact homogeneous space.
M
is equivariantly, isometrically immersed in
M M
vectors
iRm,
k-type with k s m.
is of
Proof.
m+ 1
Let
Let H,
u
be an arbitrary point of i
at
u
M.
Then the
are linearly dependent.
259
§ 2. Submanifolds of Finite Type
Thus, there is a polynomial P(6)H = 0
at
immersed in
u.
IRm,
P(t) = 0
with k s m.
M
Because
P(A)H = 0
by Theorem 2.1 we see that because
P(t)
M
has at most
(Q.E.D.)
of degree s m
such that
is equivariantly isometrically at every point of
M.
is of finite type.
k
roots,
M
is of
Thus,
Moreover,
k-type
6. Submanifolds of Finite Type
260
Examples of
03.
2-type Submanifolds
According to Proposition 2.1 of Takahashi, minimal submanifolds of hyperspheres of of
Rm.
are
R1Q
1-type submanifolds
Moreover, Corollary 2.1 shows that there exist many
important finite type submanifolds in Rm .
In this section,
2-type submanifolds in Rm
we will give many examples of
(Chen [22,25]) . Let
Example 3.1, (Product Submanifolds).
and Rm .
be two compact submanifolds of Rm M xM
Then the product submanifold only if both
M
both M and M M xM
and
M
are of
Moreover, if
For instance, con-
2-type.
Consider the isometric imbedding
x = x(r,s) _ (a cos
a,
is the lattice
A
(0,2rrb). Then
and
is isometric
T2
T2 = S1(a) xS1(b).
to the product of two plane circles;
(3.1)
respectively.
1-type, then the product submanifold
sider the flat torus T2 = R2 /A, where
(2rra,O)
and M
is of finite type if and
are of finite type.
is either of 1-type or of
generated by
M
x
of
T2
in R4
by
a sin a, b cos b, b sin b)
Then, by direct computation, we find
-(.+ -7) , ar as 2
(3.2)
(3.3)
(3.4)
H
2
x0 = (0,0,0,0)
I(acosa, asina,
AH = 7(- .cos S, a
1 a
fibSsinb) cos.
sing,
-1
b
cos b,
-
'sin b)
b
§ 3. Examples of 2-type Submanifolds
(3.5)
a
261
as ab bb
2
H =T 1(1a cos r, 1 sin r, 1 cos e, 1 sin s) 16
if and only if
Assume that
a = b.
1-type in
is of
T2
From (3.1) and (3.3). we see that
II24
From (3.3),
a 9d b.
(3.4) and (3.5), we obtain (3.6)
-1.7+ -1.f) AH + -.T1,sl
a2H
Thus, by Theorem 2.2,
a
b
a T2
is of
P(t) - t2
(3.7) Then
P(t)
(1,2)
Example 3.2.
in
1R4
a2;17
Thus, by using Proposi-
-17.
b tion 3.5.7, we can conclude that if a-7
is of order
Let
2-type.
a1 +b1)t+ and
has roots
=O
a > b >
then
T2
.
(A flat torus in
Again consider
1R6 .)
the flat torus
T2 =
(3.8)
with
A
generated by
IIt2 /A
,
Let x : T2 _. IIt6
((2Tra,O) , (0,2Trb)) .
be defined by
(3.9)
x = x(s.t) = (a sins, bsin ssint, bsinscosS, a cos s, b cos s sins, b cos s cosh)
Assume that
(3.10)
a2 +b 2= 1
and
a,b > 0
.
6. Submanifolds of Finite Type
262
By a direct computation, we have
(3.11)
H=
(O, sins sin b, sins cos., O, cos s sin b ,
+
cos s cos
b
)
,
SH = (1 + - )H - a (sin s, O. O, cos s, O, O)
(3.12)
2b
b
(3.13)
= (1 +) 2H -(2
A
b
2b
+1b ) (sin s, O, O, cos s, O, O )
Consequently, we have
(3.14)
A2H
- (2 +
) GH + (1 + 2)H = 0
This shows that
T2
is of
2-type in
Example 3.3.
and x :M a ]Rm
IIt6
(Diagonal immersions.)
x :M -4 IRm
be two isometric immersions of a compact
Riemannian manifold M
into
and
]Rm
Then the normalized diagonal immersion defined by
Let
x'(p) = 1 (x(p),x(p))
respectively.
ltm,
x' :M + IItM+m
is of finite type if
2
and x are of finite type.
and only if both
x
ular, if both
and x are of
that
x'
x
is either of
1-type, then we can show
1-type or of
For example, consider the unit
(3.15)
S2 =
2-type.
2-sphere in
( (x,y,z) E IIt3 1 x2+ y2+ z2 = 1)
Define an isometric immersion
In partic-
u :S2 + 7R8
by
1R3
.
by
.
§ 3. Examples of 2-type Submanifolds
ul =
u2 =
,
(3.16)
u 4 = YE
- y2)
(x2
2
u6
v2
5
u7
u3 =
Y
263
,
2
u8 = 17 (x2 + y2 - 2z2)
,
2
Then, by a direct computation, we can see that
order
[1.2]
in
Example 3.4.
Ilt8
Thus,
.
S2
is of
is of 2-type in
S2
IIt8
H(2n +2; C)). Let S4n+3 C2n+2 = S4n+4 given by denote the unit hypersphere in S4n+3
=
(MM,n
in
2n+1
((z0,...,z2n+1)t E C2n+2
IzAI2
= 1)
A O In
S4n+3
we have the following generalized Clifford torus
M2n+1, 2n+1 = S
2n+1
1 (
)xS
2n+1
1
(
)
defined by
M2n+1,2n+1
(3.17)
2n+l
{(z0,...,z2n+1)tEC2n+2 It n IztI2
O Let
GC = (z E C (IzI = 1).
acting on
S4n+3
and on
Denote the quotient space
Mn,n
Then
.
is a group of isometries
GC
M 2n+1,2n+1
; t=z1 Izt'2 .)
by multiplication.
M2n+1,2n+1 /GC
by
Mn.n.
Then
admits a canonical Riemannian structure such that C
M2n+1,2n+1
> Mn,n
6. Submanifolds of Finite Type
264
becomes a Riemannian submersion with totally geodesic fibres S1
Moreover, we have the following commutative diagram:
.
1
M2n+1,2n+1
) S
4n+3
(3.18) QP2n+1
MT n,n
where
i
and
are inclusions.
i'
minimal in S4'3, Mn,n of Cp2n+1 Let
cp
:TP2n+1
standard imbedding of
Since
M2n+1,2n+1
is
is a minimal (real) hypersurface
denote the first
> H(2n +2; T) cP2n+1
into H(2n +2; C defined
by (cf. *4.6) ip(z) = zz*
(3.19) Then
:p
induces an isometric imbedding of
H(2n + 2; C).
of
Mean
(3.20)
Because,
Mean
into
By a direct long computation, we may prove
that, for any point
H
.
in
H(2n + 2; T)
H=
the mean curvature vector
A E cp(Mn,n),
T
AA - -(4n+ 1)H,
at
A
is given by
(2I - (4n+3)A -At)
(3.20) implies
(3.21)
AA = 2(4n+3)A.+2At-4I
,
(3.22)
AAt = 2(4n+3)At+2A-4I
.
§ 3. Examples of 2-type Submanifolds
265
(3.21) and (3.22) we may obtain
From (3.20),
P(A)H = 0 ,
(3.23)
P(t) = (t - 4(2n+ 1) ) (t - 4(2n+ 2)) . Consequently, by applying Theorem 2.2, we obtain the following
where
Proposition 3.1. H(2n+2; Q).
Moreover,
Example 3.5. unit hypersphere
is a 2-type submanifold of
Mn,n
4(2n+1), r(2n+2) E Spec (M11
(MQ,n
in in
S8n+7
H(2n +2; Q)). Q2n+2 = S8n+8
have the generalized Clifford torus
Consider the we
S8n+7
In
M4n+3,4n+3
defined by
M4n+3,4n+3 =
2n+1
((z0,...,z2n+1)tEb2n+21 nE Iz1,2= GQ = fz E Q Ijzi = 1).
Let
acting on
S
Bn+7
and on
GQ
M4n+3,4n+3
Denote the quotient space MQ
The
M
4n+3,4n+3
by multiplication. by y
/ GQ
given by
rp(z) = zz*.
n,n
Then
QP2n+1
Consider the first standard imbedding into H(2n + 2; Q)
)z12=)
is a group of isometrics
is a minimal real hypersurface of
n
1
j=n+l
i=O
cp
of
QP2n+1
Then, by a long
direct computation, we can prove that the mean curvature
vector
H
of
MQ,n
n
in H(2n+2; Q)
at
A E cp(MnQ,n)
given by
(3.24)
H=
8n+3 (21 - (8n + 7)A -At)
is
6. Submanifolds of Finite 7)'pe
266
Since
AA =
- (8n+3)H,
(3.24) implies P(A)H = 0
(3.25)
,
P(t) _ (t-4(4n+3))(t-16(t+l)).
where
Consequently, by
applying Theorem 2.2, we have the following. MQ,n
Proposition 3.2.
2-type submanifold in
is a
H(2n+2; Q) . Example 3.6.
(MQ,n,n
following product of three
in
H(3n +3; Q)).
Consider the Q3n+3
(4n +3)-spheres in
defined in an obvious way;
_ 4n+3
M=M4 +3 4 n+3 4 n n , ,
+3-S
1 1 1 (-) xS 4n+3 (-) xS4n+3 (-) 13
V3 CS12n+11(1) CQ3n+3
Then GO = (z E Q (Izi = 1)
in
2
QP
3n+2
Then Qp3n+2
12n+ll (1)
and on
M / GQ
by
MQ,n,n
Then
into
cp
induces an isometric
H(3n + 3; Q).
By a long computation, we may prove that the mean curvature vector A E
(3.26)
cP(M4,n,n)
H=
M
is a minimal submanifold of codimension MQ,n,n Consider the first standard imbedding cP of
into H (3n +3; Q)
imbedding of
S
Denote the quotient space
by multiplication. MQ non.
acts on
H
of
MQ,n,n
in
H(3n +3; Q)
at
is given by
n+
(32I - 96 (n + 1) A + 21 (A -At) 1
§ 3. Examples of 2-type Submanifolds
267
this implies P(n)H = 0, where P(t) = (t -24n - T) (t -24n - 24) . Consequently, by Theorem AA = -6(2n +1)H,
Because
2.2, we obtain the following. Proposition 3.3.
MQ
2-type submanifold in
is a
n,,
H(3n+3; Q). Example 3 .7 .
in H (n + 2; V). Let Cpn+l be
(Qn
the complex projective
(n + 1)-space with constant holomor-
phic sectional curvature geneous coordinates of
4.
z0....,zn+l
Let
be the homo-
Then the complex quadric
CPn+1
Qn
is defined by
n+I E Jzi12
Qn = ((z0,...,zn+l) E CPn+1
=
01
i=O Denote by
cp
H(n+ 2; (r).
the first standard imbedding of
A E cp (Qn)
(3.27)
Thus we have
into
Then, by a direct computation, we may prove Qn
that the mean curvature vector of
a point
CPn+i
in
H(n + 2; C)
at
is given by H=
n(I - (n+l)A -At)
P(A)H = 0,
where
P(t) = (t-4n)(t-4(n+2)).
Therefore, by applying Theorem 2.2, we have the following.
Proposition 3.4 (A. Ros manifold in H (n + 2; C) . Example 3 .8 .
(MI n
[ 2 ]) .
Qn
is a
in H (2n + 2, ]R))
following generalized Clifford torus in
.
2n+1
2-type sub-
Consider the S.
6. Submanifolds of Finite Type
268
Sn(1) xsn(
Mn,n =
V2
a S2n+1(1) C IR2n+2
/2
Denote by
defined in an obvious way.
G
the group of
isometries generated by the antipodal map. the quotient space
Mn,n /G.
Einstein hypersurface of
Then nM n
cp
H (2n + 2:
]RP2n+l
induces an isometric imbedding of Ilt)
.
MIR n
is a minimal
1RP?n+l = S2n+1 / G
the first standard imbedding of Then
Denote by
.
Denote by
cp
into H(2n+ 2; ]R) Mm n n
into
By a long computation as before, we many prove
that Mn n is a Remark 3.1.
2-type submanif old in H (2n + 2; ]R) . Although examples given in this section are
spherical, there exist some finite-type submanifolds which are not spherical.
(cf. Remarks 5.3 and 5.4.)
§ 4. CAaracterizations of 2-type Submanifolds
269
Characterizations of 2-type Submanifolds
44.
In this section. we will give some characterizations of subIn order to do so, we need to recall the
manifolds of 2-type.
definition of allied mean curvature vector introduced in Chen [7) and to compute
t H.
M be an n-dimensional submanifold of an m-dimensional
Let
Riemannian manifold
en+1" ",em be mutually orthogonal
Let
N.
M
unit normal vector fields of
N
in
parallel to the mean curvature vector a normal vector field
a(H)
such that H
of
M
en+l in
N.
is
We define
by
m
a (H) = E tr (AH Ar) er.
(4. 1)
r=n+2
Then
M in
is a well-defined normal vector field (up to sign) of
a(H)
in
N.
N.
We call
a(H)
It is clear that
Definition 4.1. N
the allied mean curvature vector of M a(H)
is perpendicular to
in
Remark 4.1.
N
of a Riemannian manifold
if the allied mean curvature
N
is called an Q-submanifold of
vector of M
M
A submanifold
H.
vanishes identically.
For results on a-submanifolds, see for instance,
Chen [7), Houh [1], Rouxel [1), and Gheysens, Verheyen, and Verstraelen [1,2). M
Let
vector
(4.2)
H.
be a compact submanifold of Rm with mean curvature For a fixed vector
c
in Rm
fc = < H, C >.
we put
6. Submanlfolds of Finite Type
270
Then, for any tangent vector
of
X
we have
M.
Xfc = -
(4.3)
tangent to
X, Y
Thus, for vector fields
we find
M,
YXfc = -
(4.4)
X
Thus, we obtain
n
n
A
(4.5)
Ei)
(vE
i=l
i=1
i Dn
< 6 H , c > + E <(VE. AH) E. + AD i=1
where
E1,...,E
1
Ei
M
is an orthonormal basis of
Ei + h(EiAHEi),c> and
AD
the
Laplacian of the normal bundle, that is,
n
(4.6)
ADH
= E (DV i=1
Because (4.5) holds for any
Ei
c
E
Regard
v AH and
ADH
i Ei
in
AH = ADH + E (h(Ei,AHEi)
(4.7)
H - DE D
i
(4.5) implies
Htm,
+ AD
Ei
H).
H
E
i + (VE 1AH) Ei).
as (1,2)-tensors in
T M 0 T M 0 TM
defined by
(4.8)
(v AH) (X, Y) = (VX AH) Y ,
(ADH) (X, Y) = ADX H Y.
We put
(4.9)
V AH = V AH + ADH.
§ 4. Characterizations of 2-type Submanyolds
271
Then we have
n
tr (v AH)
(4.10)
We notice that if Let
(vE AH) Ei) . i Ei H Ei +
E (AD
i=1
we have
DH = 0,
En+l,...,em en+l
such that
32m
=
be an orthonormal normal basis of is parallel to
E h (Ei,AH Ei)
(4.11)
II An+1 II2 = tr (A2n+l )
where
v AH = V AH.
.
=
II An+l
H.
112
M
in
Then we have
H + a(H) ,
Combining (4. 7) , (4.10), and (4-11),
we obtain
Lemma 4.1.
Let
M
Htm.
be an n-dimensional submanifold of
Then we have
AH = CDH + IIAn+lII2H + a(H) + tr(vAH).
(4.12)
For the comparison with 2-type submanifolds, we give the following
Let
Proposition 4.1.
M
be a compact submanifold of
Ht m
If M has Parallel mean curvature in )Rm, then M is of 1-type if and only if (1) II An+1 II2 is constant, (2) tr (v AH) = 0, and is an C!-submanifold of I.
(3) M
Proof. If
M
DH = 0,
is of 1-type in
AH = b H.
(4.13)
Because
H2m,
we have
AD H = 0
there is a constant
Thus, by Lemma 4.1, we have
IIAn+l II2H + a(H) + tr(VAH) = bH.
and
b
v AH = v AH.
such that
6. Submanlfolds of Finite Type
272
H. a(H),
Since
and
are mutually orthogonal, we obtain
tr(v AH)
(1). (2). and (3) of the proposition. Conversely, if (1), (2), and (3) hold, then, by setting b = I I An+l II2, we obtain A H = b H. Thus, by Theorem 2.2, we conclude that M M
Now, we assume that manifold of a hypersphere
Rm.
at the origin of vectors of
M
Rm
in
(Q.E.D.)
is an n-dimensional compact subof radius
Sm-1(r)
Denote by and
is of 1-type.
H
and
H'
in Rm
r
the mean curvature
respectively.
Sm-1(r),
centered
Then we have
(4.14) where
x
be the unit vector parallel to where
M
denotes the position vector of H'.
in
Rm.
Then we have
H' = a'
We choose an orthonormal normal basis
a' = IH'1.
en+1' -,em of M in Rm such that (4.15)
Let
en+l = H / a , en+2 = ( + a' x) / r a,
where 1
(4.16)
a = (HI = (a'2 +
Because Ax = - I,
we have
(4.17)
tr (AH An+2)
(4.18)
tr (AH Ar)
r2
)2
= a ' (IIA9112 -n (a') 2) / r a, = tr (AH , Ar) , r = n+3,. .. , m.
From these, we obtain
P
§ 4. Otaracterizations of 2-type Submanifolds
a (H) = a' (H') + r a'
(4.19) where
n (a')2 ) en+2' M
denotes the allied mean curvature vector of
a'(H')
Sm-1 (r).
in
For the normal vector field x
273
we have
x,
M
is parallel in the normal bundle of
any normal vector < DTI,x > = 0.
D'
where
in
M
Thus, for
Rm.
in
i.e.,
with < x, fl > = 0,
Htm
we have
From these, we find that aD
(4.20)
of
Tj
Dx = 0,
H=
QD '
H'
is perpendicular to x,
denotes the normal connection of M
in
Sm-1(r).
From (4.14) and (4.15), we also find (4. 21) a2 IIA
n+lH !12
= tr (A , + I 2) 2
=
r
(a )
2
IIAS II
2 +
2n (a") 2 + n
r2
r4
Therefore, by combining'(4.12), (4.14), (4.19), (4.20), and (4.21), we obtain the following. Lemma 4.2.
hypersphere
(4.22)
Let
Sm-1 (r)
M be an n-dimensional submanifold of a of radius
r
in
Mm.
Then we have
A H = AD' H' + a ' (H') n++ tr (v AH) + «' (IIAtII 2 + 2)
-
-r2 (x-c0), where
c0
denotes the center of
Sm-1(r).
We need the following. Definition 4.2.
metrically imbedded in
Let M be a symmetric space which is iso-
Rm
by its first standard imbedding.
Then
6. Submanifolds of Finite Type
274
of M
M
a submanifold
centroid (i.e., the center of mass) of
of
M
is called mass-symmetric in
M
Rm
in
if the
is the centroid
in Rm.
M
Let
Lemma 4.3.
hypersphere symmetric in Proof.
M be a compact minimal submanifold of a of radius
Sm(r)
Rm+1
in
r
M
Then
is mass-
Sm(r).
Because
A x = - n H ,
Hopf's Lemma implies
HdV=0. M
M
Since
is minimal in
Sm(r),
we have
H =
12
(c - x),
where
r
c
is the center of
Sm(r)
Thus we find
Rm+1.
in
c = f x dV / f dV. M
This shows that
M
is the centroid of M
c
in
Rm+1
(Q. E. D.)
Lemma 4.3 shows that compact minimal submanifolds of hyperspheres are special examples of mass-symmetric submanifolds.
In
fact, there are many mass-symmetric submanifolds which are not miminal submanifolds of a hypersphere (Cf. Examples 3.1-3.8). By using Lemma 4.2. we have the following. Theorem 4.1.
(Chen (251.)
Let
pact, mass-symmetric submanifold of
type in
]Rm,
Sm-1(r).
If
M
is of 2-
then
(1) the mean curvature
and is given by
M be an-n-dimensional, com-
cx'
of
M
in
Sm-1 (r)
is constant
275
§ 4. Characterizations of 2-type Submanifolds
(a')2
(4.23)
=
(n)2 ( 2 - Xp) (aq
r
(2)
tr (v'A H.) = 0,
(3)
ADH' + a'(H')
and +
(jjAj!l2 + 2) H'
r
denotes the Weingarten map of
A'
where
- 2) r
VA 'H' + AD'
=
M
(),p + Xq) H', in
'H1.
Conversely, if (1), (2), and (3) hold, then in
center of
is of 2-type
Without loss of generality, we may assume that the Sm-1 (r)
Rm,
(4.24)
is the origin of
Eim.
If M
is of 2-type
then Theorem 2.2' and Lemma 4.3 imply
AD H' + a'(H') + tr (v AH) + a' (UAgEl2 -
n r
+ bH' for some constants M
M
Rm Proof.
in
and
Sm-1(r)
b
and other terms in
and
r2 c.
2
S - n2 r
x - x = 0, n Since
tr(v A H)
(4.24) are normal to
M,
is tangent to we have
On the other hand, because A H = A . + 2 I r we have tr (v'A H.) = 0. Furthermore, because
tr (v A H) = 0. DH = D'H',
normal to Sm-1(r) and other terms in (4.24) are tangent to Sm-1(r), we obtain from (4.24) that (4.25)
(a')2
On the other hand,
(4.26)
+ r12 = a2 = - nb - ncr2 (4.24) gives
P(s) (x) = 0.
and x
is
276
6. Submanifolds of Finite Type
where
P(t) = t2 + bt - n.
[p,q],
M
Since
is of 2-type with order
(4.26) implies b = - (lip + Xq)
and
c = Xp Xq.
Statement (3) follows
by (4.25), we obtain Statement (1).
and equation (4.24).
from Statements (1) and (2)
The converse
of this follows from Theorem 2.2' and Lemma 4.3.
If M is a hypersurface of
Thus,
(Q.E.D.)
then we have the
Sm-1(r),
following.
Theorem 4.2.
(Chen [25].)
symmetric hypersurface of
Rm+l
M be a compact, mass-
Let
Sn+1(r).
If
M
is of 2-type in
then (1)
the mean curvature
a of M
in
Iltn+2
is constant
and is given by
(4.27)
(Xp + ),q) - (n)2 lp lq o
a2 =
n (2)
the scalar curvature
(4.28)
n An+2
of M
(lp + ) q) - n -1) (nr
T=
(3)
T
is constant and is given
ap aq
the length of the second fundamental form
h
of M
in
is constant and is given by
(4.29) (4)
IIh!I =
and
tr(VAH) = 0.
Proof.
Sn+l(r).
ap + Xq ,
Let
If M
M be a compact mass-symmetric hypersurface of is of 2-type, then Theorem 4.1 implies that the
§ 4. Characterizations of 2-type Submanifolds
mean curvature
of
a'
M
M
Since the codimension of vature vector that is,
(a')2 A 'H,
+
H'
M
of
r2
=A
H
tr(v A H) = 0.
we also have
is a non-zero constant.
Sn+l (r)
is therefore parallel,
Sn+1(r)
DD H' = 0
is one, the mean cur-
too.
Because,
a2 =
equation (4.23) implies (4.27). Since we have
,
-
in
in
Thus
D'H' = 0.
Sn+l (r)
in
277
2 I,
statement (2) of Theorem 4.1 implies
r
M
Now, because a'(H') = 0.
is a hypersurface of
Sn+1(r),
Thus, by statement (3) of Theorem 4.1,
we obtain
(4.30)
IIAg112 +
Because
2r
11h112 = 11Aj112 + 2 ,
r
= lp + aq.
(4.30) implies (4.29). Equation
(4.28) follows easily from equations (5.2.30),
(4.27), and
(4.29).
(Q. E. D. )
As a converse to Theorem 4.2, we have the following. Theorem 4.3.
(Chen 125].)
symmetric hypersurface of curvature
a
tr(v A H) = 0, Proof.
center of and
T
Let M be a compact mass-
Sn+1(r).
If
M
has constant mean
and has constant scalar curvature then
M
'r,
and if
is either of 1-type or of 2-type.
without loss of generality, we may assume that the Sn+1 (r)
is the origin of R
are constants.
by Lemma 4.2, we find
Then we have
n+2
DD H'
Assume that = a'(H')
= 0.
a Thus,
278
6. Submanifolds of Finite Type
O H = a'( IIA
(4.31)
2
II2 + 2)
nr 2 x
r
= .'11h Since
and
a
r
Because
stants.
r
are constant,
a'
H = a'g - x/r2,
A H - 11h 112 H +
(4.32)
- nag X.
112 C
112
and
(4.31) implies
x = O.
(na2-IIhII2)
rZ
are also con-
11h
Consequently, by applying Theorem 2.2', we see that M
is either
of 1-type or of 2-type.
(Q.E.D.)
As a special case of Theorem 4.1, we also have the following. (Chen [25].)
Theorem 4.4.
M be a compact. mass-symmetric submani-
j,g,
fold of a hypersurface
parallel mean curvature vector
2-type if and only if (1)
H'
IIA H II
in
Proof.
Let
M
If
Sm-1(r),
is constant,
is an Q-submanifold of Sm-1 (r)
and (3) M
Sm-1(r)
Rm.
of
Sm-1 (r)
has non-zero
M
then
is of
(2) tr (V A H .) = 0,
.
M be a compact mass-symmetric submanifold of M
such that
is of 2-type.
zero parallel mean curvature vector
Assume that H'
in
M
has non-
Sm-1(r).
Then, by
Theorem 4.1, we have
(4.33) Because Since A
H
IIASII2 + '
r2
a'
a' (H') = 0.
is constant, this implies that
a'(H') = 0,
= A'H ,
= ap + lq,
+ I/r2
M
and
implies tr(VAH,) = 0.
is an Q-submanifold of
D'H' = 0,
IIA H.II
is constant.
Sm-1(r).
Because
statement (2) of Theorem 4.1
§ 4. Characterizations of 2-type Submanifolds
279
Conversely, if IIA H, H is constant, tr (V A H ) = 0 and a' (H') = 0, then, we have tr (v A H) = 0. And moreover, by Lemma 4.2, we also have
(4.34)
AH = a'(IlAsII2 + a'(IIAsll2 + r2 )
where
H = a'C - x/r2,
(4.35)
AH-
Since
a2 = (a')2 +
and
Because
(IIASII2
r2)
6
-
n
r
2r
2
X.
are constants.
(4.34) implies
+2)H+ r
r2
(na2
- CIA;II2 - 2) x = 0.
r
(4.35) and Theorem 2.2' imply that M
H' # 0,
2-type in Rm.
is of (Q. E. D.)
For surfaces in
S3(r),
we have the following classification
theorem (Chen [25].)
Let M
Theorem 4.5.
of
S3 (r)
in
R4
be a compact, mass-symmetric surface
Then M
is of 2-type if and only if
M
is
the product of two plane circles of different radii, that is,
M = S1 (a) X S1 (b) , Proof.
hypersphere
Let
a 71 b. M be a compact mass-symmetric surface of a
S3(r)
in
3R4.
assume that the center of
without loss of generality, we may S3(r)
is the origin of
R4.
If M
is the product of two plane circles of different radii, then, by Example 3.1, we see that Conversely, if
M
M
is of 2-type in
is of 2-type in
3R4,
3R4.
then by Theorem 4.2,
M has constant mean curvature and constant scalar curvature. Moreover, we also have
6. Subma :ifolds of Finite Type
280
(4.36)
tr(7AH') = 0.
by virtue of D'H' = 0.
Let
El, E2 be the eigenvectors of AH
Then we have (4.37) where T
i = 1, 2,
A H , Ei = .1i Ei, _l, -2
are the eigenvalues of
are constants,
.-l,
-2
Because,
A H
and
are constants.
We pu t
(4.38)
2
V E1 =
E..
J
j=1
3
Then we find
(vE1 AH,) E1 = (-
(4.39)
wl2 (E1) E2
Similarly, we also have
vE AH) E2
(4.40)
2-..1) w2 (E2) E1.
2
Because obtain
M
u
is of 2-type, = 0.
Thus, by
-2.
tr(v A H,) = 0, we
From these, we may conclude that
the product of two plane circles.
Because,
M
M
is in fact
is of 2-type, the
radii of these two plane circles must be different. Remark 4.2.
In general, if
with A H E i = M E .
,
only if
(4.41)
Eµ= i i jji
Remark 4.3.
M
i = 1, ... , n,
iji
(u-µ) w(Ei)
is a submanifold of
(Q.E.D.)
Sm-1(r)
then tr (17 A H ) = 0 if and
1.... ,n.
Recently, A. Ros [2,3] has applied the concept
of 2-t rpe SubutaniJolds
4.
281
of order and the spectral decomposition (1.10) introduced in Chen [15, 17. 22) to obtain some further results concerning 2-type submanifolds which we shall mention as follows: Let
.
be a minimal isometric immersion
Sm-1(r)
: M -4
of an n-dimensional, compact. Riemannian manifold IItm
centered at
the Euclidean coordinates of
X1,.... xm
into a
Denote by
0.
Sm-1(r)
in
be the row matrix given by
(xi .Oxm)
x =
Let
of
Sm-1(r)
hypersphere
M
IItm.
xl,...Oxm.
m-1
(r) into H (m ; ]R) Define an isometric immersion f of S Then f is an order 2 immersion of Sm-1 (r) by f (x) = x t x . H(m
into
M
An isometric immersion of
IR).
in
is
Sm-1(r)
is not contained in any totally geodesic
called full if
M
submanifold of
Sm-1(r).
The results obtained by A. Ros [2,3)
are the following. Let
Theorem 4.6.
:M -4 Sm-1 (r)
of a compact Riemannian manifold immersion M
tr (A A')
Sm-1(r),
of
in
M
where
is a constant and
Sm-1 (r)
Let
M
A'
is Einsteinian
S,
of
M
in
is the Weingarten map
be a compact, Kaehler submanifold of
such that the immersion is full.
M
fo$ of
.
standard imbedding of
Then
M
is of 2-type if and only if
k
such that the
Sm-1(r)
= kg(,") for all normal vectors
Theorem 4.7. CPm
into
Then the immersion
is full and minimal.
H(m ;IR)
into
and
r
M
be an isometric immersion
SPm
is of 2-type in
into
Denote by
H(m + 1 ;C)
H (m +1 ;C)
Einsteinian and the Weingarten map of
cp
defined in §4.6.
if and only if M
in
the first
CPm
M
is
satisfies
6. Subnwnifolds of Finite Type
282
tr (A9'S A')
= k g (S , rl)
for all normal vectors
S, -n
of M in
The idea of the proofs of these two results is to express G H
in terms of the Ricci tensor and the Weingarten map of
M.
£Pm,
§ 5. Closed Curves of Finite Type ¢5.
283
Closed Curves of Finite Type In this section we shall study closed curves of finite
Im
type in
In order to do so, we first recall the Fourier
.
series expansion of a periodic function. Let
Then
2'rr.
be a periodic continuous function with period
f(s)
has a Fourier series expansion given by
f(s)
a
f(s) _ -2+a1 cos(y) +a2 cos(2r) +
+b1 sin(r) +b2 sin(2r ) + where
and
ak
are the Fourier coefficients given by
bk
(5.2)
ak =
T
(5.3)
bk =
-r
pTr r J
---r
pTr r
J
f(s) cos (ki)ds,
k = 0,1,2,
f(s) sin (ks)ds,
k = 1,2,---
-Try
In terms of Fourier series expansion, we have the following
(Chen [221) Theorem 5.1. Then
C
Let
be a closed smooth curve in
C
is of finite type if and only if the Fourier series
expansion of each coordinate function
xA
of
C
has only
finite nonzero terms. Proof. IItm
]Rm
Assume that
C
such that the length of
the arc length of
C.
We put
is a closed smooth curve in C
is
2Trr. .
Denote by s
284
6. Subinanifolds of Finite Type
(5.4)
x(j) =
d]x ds]
Because
- d2 - in this case, we have
0 =
ds
A H = (-1)jx(2j+2),
(5.5)
If
C
is of finite type in
j =
IRm,
0,1,2,...
then Theorem 2.1
implies that each Euclidean coordinate function in
IRm
xA
of
C
satisfies the following homogeneous ordinary differ-
ential equation with constant coefficients:
x(2k+2)
(5.6)
A
for some integer
+c1x(2k) + ...+ckx(2) A A =0
k z I
and some constants
cl,...,ck.
Because the solutions of (5.6) are periodic with period each solution
xA
2-r,
is a finite linear combination of the
following particular solutions: n.s 1, cos( r )
(5.7)
Therefore, each
(5.8)
xA
qA
x = c., + E
t= pA
,
PA, q
function
xA
A
;
,
ni,mi E a
is of the following forms:
a_ (t) cos (ts) +b_ (t) sin (ts) r 1
for some suitable constants integers
m.s sin( r )
aA(tl, bA(t), cA
A = 1,...,m.
and some positive
Therefore, each coordinate
has a Fourier series expansion which has only
finite nonzero terms.
285
§ S. Closed Curves of Finite Type
Conversely, if each
has a Fourier series expansion
xA
which has only finite nonzero terms, then the position vector x
C
of
in
takes the following form:
]Rm
q
x = c+ E {at cos(ts) +bt sin(tr)
(5.9)
t=p
for some constant vectors a, bt, c
in
and some
IRm
2
Since A = - - ,
p, q.
integers
(5.9) implies
ds q
2
Ax = E
(5.10)
(-xt,)
t=p
(at cos(ts) +bt sin(ts) )
Let xt = at cos (ts) +bt sin(ts) . x = c+ Et
that x
for
type.
c
in
p
Itm
xt
is in fact the spectral decomposition of Since
.
Then (5.9) and (5.10) show
q
is finite,
c
is of finite
(Q .E .D . )
From the proof of Theorem 5.1, we obtain the following.
Let
Corollary 5.1. in
2Trr
IItm.
If
C
be a closed curve of length
C
is of finite type, then we have the
following spectral decomposition:
(5.11)
q
x = x0+ E xt
xt = at cos(ts) +bt sin( rs)
t=p
for some vectors
at, bt
in
IRm
and some integer
p, q 2 1.
Using Corollary 5.1, we have the following (Chen (251)
Let
C
be a closed curve in
k-type, then
C
lies in a linear
Proposition 5.1. If
C
is of
]RA1
O-subspace
286
IItS
6. Submanifolds of Finite Type
of
IItm
Since
Proof. of
of
C
Let
IR2k .
IRm.
Then we have
E t2(IatI2 + IbtI2] = 2r2
E
t+t =k
tt'(
E
t+t =k
E
t-t =k
where
1 1 k 3 2q,
t-tE=k
tt'(
tt' +
for
there is a
t. p
+2 (5.14)
S-subspace
be a closed curve of length
C
is of finite type in
(5.12)
(5.13)
k,
k-type which lies fully in
Proposition 5.2.
is con-
xt
(Q.E.D.)
For each positive integer
closed curve of
If
Since each
must lie in a linear
s 1 2k.
with
Remark 5.1.
2rrr.
C
Span(at,bt1, ]Rm
k-type, there exist exactly k
is of
C
which are nonzero.
xp,...,xq
tained in
IIt8
with S s 2k.
tt'(
at,bt; p s t 9 q
are vectors in
m IIt
given by (5.11).
Conversely, if there exist
at,bt; p 3 t s q,
such that (5.12), (5.13) and (5.14) hold then
x(s) = Et
p
(at cos(ts) +bt sin(tr))
type closed curve in
)Rm.
for
in
IRm
1 s k s 2q,
defines a finite
§ 5. Closed Curves of Finite 7Ype
287
From (5.11) we have
Proof.
q
x(s) = xo+ E [at cos (tr) +bt sin (tr )
(5.15)
t= P
Thus we find from
q
= 1
.
that ,
= t,E
r2
(5.16)
< x'(s),x'(s) >
1
( < atat, > sin (r s) sin (trs) =p
+ cos (i)s cos (trs)
- 2 < at.bt, > sin (rS) cos (trs) l From this we find
2r2
(5.17)
=
t E, (
- 2
rt/)s+sin
1, cos(y), sin(r) ,...,cos(). 2sin(?-q-)
are inde-
pendent, (5.17) implies (5.12), (5.13) and (5.14). verse of this follows from Theorem 5.1.
The con-
(Q.E.D.)
Using Proposition 5.2, one may classify closed curves of finite type.
Theorem 5.2. If
C
of
IRm
(5.18)
is of ,
C
Let
C
2-type in
be a closed curve of length IItm
,
2irr.
then, up to a Euclidean motion
takes the following form:
x (s) = (a cos (
)
, a sin
13 cos (v) , 13 sin (as) , O, ... O)
6. Submanifolds of Finite Type
288
where
a
and
are nonzero constants such that
S
(pa)2+
(qB)2 = r2.
Proof.
If
is of
C
2-type in
3Rm
,
then by Proposition
5.2, we have
2r2
(5.18)
=
P2(Iap12+ lbp12) +g2(IagI2+ IbgI2)
(5.19)
IapI = lbpI.
(5.20)
a p,bp,aq,bq
IagI = lbgI
are orthogonal
,
,
.
Thus, by (5.18), (5.19) and (5.20) we obtain the theorem.
(Q.E .D. ) Remark 5.2.
bt, p s t s q if
if of
C
From (5.20) we see that the vectors
are orthogonal if k-type with
k ? 3,
are not orthogonal in general. closed curve in
iR6
is of
C
is of
then
2-type.
at,
However,
at, bt, p s t s q,
For example, the following
3-type but # 0.
(1cos2s+1cos3s, 1sin2s+3sin3s, cos s-2 cos 2s 15
sin s - sin 2s, 1 cos 2s, 2 Remark 5.3.
sin 2s) 2
In views of Proposition 5.1 and Theorem
5.2, it is interesting to give the following closed curve in
]R3
of
3-type:
289
§ 5. Qosed Curves of Finite Type
x(s) _ (-3 sin (6) + cos (3) , -3 cos (6) + sin (s)
(cos (3)+sin(5)
f2
Another application of Proposition 5.2 is to give the following. Theorem 5.3 (Chen [22]).
of finite type, then
If
is of
C
C
is a closed plane curve
1-type and hence
is a
C
circle. Proof.
If
is of finite type in
C
]R2
then
,
Proposition 5.2 implies
IagI = lbgl
(5.21)
If
is of
C
plane circle. C
i 0,
b > = 0 q
.
1-type, (5.11) and (5.21) imply that If
lies fully in
is of
C 1R4
.
C
is a
2-type, Theorem 5.2 implies that
Thus, this is impossible because
C
is assumed to be a plane curve.
Now, we assume that
is of
C
k-type with
k ? 3.
From (5.21), we see that, with a suitable choice of Euclidean coordinates of
(5.22)
IIt2,
we may assume that
aq = (a,O) ,
bq = (O,a)
On the other hand, by letting
.
k = 2q -1,
(5.13) and
(5.14) of Proposition 5.2 give
(5.23)
q
q-1 >,
a
< aq bq -1>
= -< aq-1 'b q >
.
6 Submanifolds of Finite Type
290
Thus, by using (5.22) and (5.23), we see that bq-1
a
q-1
and
take the following forms:
aq-1 = (uq-l'vq-1) ,
(5.24)
bq_1 = ( -vq_l,uq_1)
From (5.22), (5.23) and (5.24) we obtain
< at.bt' = 0, t = q -1,q
latl = Ibti (5.25)
-
Now, we assume that we have
Iatl = lbtl,
(5.26)
h s t, t s q, for some
h > p.
Then, by (5.13), (5.14) and (5.26), we find
< h-1'bgl =
(5.27)
< ah-l,bq> + < aq,bh-1> =
0
From (5.22), (5.26) and (5.27), we obtain
at = (ut, v t
(5.28)
for
t = h -l,h,...,q.
for
,
bt = (-vt,ut)
Consequently, we obtain
)
h-1 = t,f = q.
0,
latl = Ibtl,
and
Therefore, by induction, we have
g 3. Uosed Curves 01 unite Type
latI = lbtI,
(5.29)
< at,bt> = 0
t x 2
for
p f,
t, 2
5 q.
291
and (5.14) we f ind E t2 < at, a2 > = 0 t-2=k
(5.30)
tt < at,b{ > = 0
E
(5.31)
t-2=k
< aq,ap > = < aq,b2 > = 0.
In particular, these imply C
k-type with
is assumed to be of
ap,aq,bp,bq
x :C
type, then
Let
IR2.
Let
This
.
be a closed curve of length
is an isometric imbedding of finite
IIt2
circle of radius Proof.
C
is a standard imbedding of
x
IIt2
(Q.E.D.)
Corollary 5.2. If
we find that
are nonzero orthogonal vectors in
is a contradiction.
2-r.
k = 3.
Because
C
onto a plane
r.
be an imbedded finite type curve in
C
Then Theorem 5.3 shows that
C
is of 1-type.
Thus
we have
x(s) = x0+ apcos
(5.32)
Moreover, we also have p2lap12
= r2.
coordinates of
bpsin
IapI = lbp1. < ap,bp > = 0
and
Thus, by a suitable choice of the Euclidean It 2
,
x(s)
takes the following form
6. Submanifolds of Finite Type
292
x(s) = P (cos () , sin
(5.33) Thus,
C
this case
is an imbedded curve if and only if C
is a circle of radius r.
Remark 5.4.
curves in
Iltm
and finite type curves in
a finite type curve in
Iltm
In
(Q.E.D.)
Theorems 5.2 and 5.3 show that both
that is, they lie in a hypersphere of
example, the
p = 1.
Iltm.
IIt2
2-type
are spherical,
But, in general,
is not necessary spherical.
For
3-type curve given in Remark 5.3 is not spherical.
§ 6. Order and Total Mean Curvature
293
Order and Total Mean Curvature
§6.
In this section, we will relate the notion of the order of submanifolds with total mean curvature.
In particular,
we will obtain a best possible lower bound and a best possible upper bound of total mean curvature. First we give the following formula of Minkowski-Hsiung n = 2; Chen (4] and Reilly [1] for
(see, Hsiung [2] for
general
n).
Let
Proposition 6.1. immersion of a compact
M
into
]Rm
.
x :M -4 JRm
be an isometric
n-dimensional Riemannian manifold
Then we have
f
(6.1)
E1+<x,H>)dV = 0 M
Proof.
Because
Ax = - nH
(Lemma 4.5.1), Proposition
3.1.4 gives
n f <x,H>dV = - (x,Cx)
(dx.dx) _ -n f
where we have used the identity
< dx,dx > = n.
Recall that for an isometric immersion an
[p,q] = [p(x),q(x)]
of
(Q.E.D.)
x :M - IRm
n-dimensional compact Riemannian manifold M
we have the order
dV M
M
M.
into
of IRm
Using the
concept of order we have the following best possible lower bound of total mean curvature (Chen [15]). Theorem 6.1.
Let
M
be an
n-dimensional compact
294
6. Submanifolds of Finite Type
submanifold of
IRm
Then we have
.
k
f
(6.2)
(-2)7 vol (M) ,
IHIkdV
,n
k = 2,3,
M
The equality holds for some only if
is of order
M
k = 2.3,...,
k,
or
n,
if and
p.
Because
Proof.
q
x = x0 + E xt t= p
n2 f IHI2dV
Axt = atxt
,
q n2(H,H) _ (AX, AX) = E atf'xt''2
=
t= p
M
On the other hand, (6.1) and (6.3) imply q
n j dV = -n(x,H) = (x,Ax) = E XtixtIl 2
(6.5)
t= p
M
Thus, by (6.4) and (6.5), we find
q
n2f IHI2dv -na f P
M
E
dV = M
Xt(at -Xp) 'xt'`2 - 0
t=p+1
Therefore, we obtain
fM
(6.6)
(n)vol(M)
IHI2dV
with equality holding if and only if
m
is of order
P.
Now, by using Holder's inequality, we find 1 %
(n
vol (M)
JM
IHI28V f
(
IF1I2rdV)r(
JM
1
pM dV)s
295
§ 6. Order and Total Mean Curvature
+s = 1, r,s > 1.
with
Let
r =
'k.
We obtain inequality
r The remaining part is clear.
(6.2).
Since we have in
Iltm
,
p it
(Q.E.D.)
for any compact submanifold M
1
Theorem 6.1 implies
submanifold of
M
Let
Theorem 6.2. ]Rm
be an
n-dimensional, compact
Then we have
.
k
(6.7)
("1)1 vol (M) ,
IH,kdV
k = 2,3,...,n
M
k, k = 2,3,...,
equality holding for some
only if
or
n
if and
is of order 1.
M
Remark 6.1.
Inequality (6.7) is essentially due to
In fact, he proved inequality (6.7) for
Reilly (2).
k = 2
by applying the minimum principle without using the concept of order.
He also proved that if the equality sign of (6.7)
holds for
k = 2,
hypersphere.
then
M
is a minimal submanifold of a
According to Theorem 6.1, we can further say
that the equality holds if and only if
M
is of order
1.
Remark 6.2. Masal'cev (1] obtained in 1976 the following result.
Let
M
be a compact orientable hypersurface of
Then
(6.8)
jjhjj2dV
f M
z 11 vol (M)
,
]Rn+1
.
6. Submanifolds of Finite Type
296
where h.
denotes the length of the second fundamental form
IIhII
The same inequality for any compact submanifold
M
in a
with arbitrary codimension was obtained independently by
IRm
Sleeker and Weiner [1] about the same time.
Moreover, Bleeker
and Weiner showed that the equality sign of (6.8) holds if and
M
only if
is in fact an ordinary hypersphere in a linear
of
n+l
(n+1)-subspace
1R
II2m
By using the notion of the orders of submanifolds we may also obtain the following best possible upper bound of total mean curvature (Chen [22]).
submanifold of
(6.9)
3Rm
n)
k 7 vol (M) ,
equality holding for some if
M
is of order Proof.
of
IItm
.
Let
both
4H
a(H) a(H)
n-dimensional, compact
k = 1,2,3, or
k, k = 1,2,3
or
if and only
4,
q.
M be an
n-dimensional compact submanifold
ADH+ IIAn+1R2H+a(H) +tr(VAH)
,
is the allied mean curvature vector. and
tr(DAH)
are perpendicular to
implies
(6.11)
4
From Lemma 4.1, we have
(6.10)
where
be an
Then we have
.
IHlkdV
J
M
Let
Theorem 6.3.
<6H,H> _
.
H,
Since (6.10)
§6. Order and Total Mean Curvature Furthermore, from (6.3),
297
(6.4) and (6.5) we also have
(6.12)
n 2 fM IHI2dV =
(6.13)
n2 f
:t=qp atllxtli2 q
< H,AH NdV = M
£, Xtllxtll2 t= P
q
n f dV = E atllxtlI2
(6.14)
t=p
M
Assume that
q < -
.
We put
A = n2 f
(6.15)
p
+ n Xpaq f dV
.
Then we have
q-1 (6.16)
_
Z'
t=p+1
(Xt - p) (at - aq) IIxtII2
with equality holding if and only if
M
0
is either of
1-type
or of 2-type. Combining (6.11),
(6.15) and (6.16), we find
n2 f < H,.DH .dV + n2 f IIAHII2dV
(6.17)
-n2(Xp+Xq) f IHI2dV+nXpaq f dV Since
M
(6.18) Let
0
is compact, Hopf's lemma implies
f < H,ADH ;dV = f IIDHII2dV
.
denote the eigenvalues of
AH.
Then it
298
6. Submanifolds of Finite Type
is easy to verify that
(6.19)
IIAH II2 =
nIHI4+n
i,j
(ki -k2
Combining (6.17), (6.18) and (6.19) and Schwartz's inequality, we get
(6.20)
0 ? n2 f IIDHII2dV+n3 f IHI4dV
+ n 1 f (ki -kj)2dV -n2(X +Xq) f IHI2dV+nXpaq f dV z n2 f IIDHII2dV + n3 ( f IH2dV) 2 /J dV
+ n3 f IHI4dV
-n2(Xp+), q) f IHI2dV+nXpaq f dV
Hence, we obtain
(6.21)
0 g n vol (M) f IIDHII2dV +vol (M) T f (k. -kj) 2dV i<j +
(n f IHI2dV - ),p vol (M) ) (n f IHI2dV
Xq vol (M) )
Combining Theorem 6.1 with (6.21), we obtain
(6.22)
f
I H 12dv
(-g) vol (M)
Substituting (6.22) into the first inequality of (6.20), we obtain
(6.23)
f IHI4dV g
2
vol (M)
By using Holder's inequality, we obtain
§ 6. Order and Total Mean Curvature
299
1k
k
f IHikdV 5 ( f IHI4dV)4 ( vol (M) ) for
k < 4.
4
Thus, by applying inequality (6.23), we obtain
inequalities (6.9) . If the equality sign of (6.9) holds for some
k,
then
all the inequalities in (6.16) through (6.22) become equalities.
Thus, we find that
umbilical.
H
is parallel and
M
is pseudo-
Hence, by applying Proposition 4.4.2 of Yano and
Chen, we conclude that
is easy to verify.
M
is of
1-type.
The remaining part
(Q.E.D.)
An immediate consequence of Theorem 6.3 is the following
(Chen [221). Theorem 6.4. manifold of
(6.4)
IRm
.
Let If
M IHI
be an
n-dimensional compact sub-
is constant, then
xp ` nIHI2 - lq
Either equality sign holds if and only if
M
is of
1-type.
300
6. Suhmanijolds of Finite Type
Some Related Inequalities
§7.
In this section, we give some geometric inequalities
which are also related to the notion of the order (Chen [22].) Proposition 7.1.
submanifold of
(7.1)
f
]Rm
.
Let
M
be an
Then we have IH12dV _ 1 2
'dH 2dV z n(XI + a2) f
M
M
equality holding if and only if Proof.
n-dimensional compact
M
is of order
< 2.
From (6.12), (6.13) and (6.14) we find
n2(SH,H) -n2(XI+?,2) (H,H) -)`1X2 f dV
(7.2)
E at(at -xi)(at -a2)IIXt112 a
t_'3
On the other hand, we also have
If the equality of (7.1)
holds, then the equality of (7.2) holds. s 2.
0
(AH,H) = (6dH,H) = (dH,dH).
Thus, from (7.2), we obtain (7.1).
order
p d JM
n
Thus,
M
is of
The converse of this follows from (7.2)
immediately.
Remark 7.1.
Ros also obtained Proposition 7.1 independ-
ently (see Ros [21). Proposition 7.2.
submanifold of
7ltm
.
Let
M
be an
n-dimensional compact
Then we have
2k+1 (7.3)
fM
I6kHII2rdV
)rvol (M)
§ 7. Some Related Inequalities
301
2k+1
k
f lidokHl!2rdv ? (1n )rvol (M)
(7.4)
M
and
r z 1
for
k = 0,1,2,...,
where
60H = H.
sign of (7.3) or (7.4) holds for some
if
M
k
if and only
is of order p. Proof.
(7.5)
and
r
The equality
Because
x = -nil,
we have
11cc
n2(6kH,6kH)
n2 f
q
=E
X21+2Ilxtll2
t= p
q
n2 f dV = -n(x,H) = E X llxtIl2 t=p
n2
J
II6kHII2dv -na2pk+1 vol (M)
q (X2k+l
t=p+l This shows (7.3) for
r = 1.
x2k+1) Xtllxtll2 p
0
By applying Holder's inequality,
we may obtain (7.3) for -r > 1.
For (7.4) we consider
q (7.8)
n2(dAkH,dAkH) = n2(AkH,Ak+1H) =
2k+3IIxtII2
t= p By using (7.6) and (7.8), we obtain (7.4) for
by applying Holder's inequality, we obtain (7.4) for The equality cases can be easily verified.
Thus,
r = 1.
r > 1.
(Q.E.D.)
From Proposition 7.2, we obtain immediately the following.
302
6. Submanifolds of Finite Type Corollary 7.1.
submanifold of
7Rm
(7.9)
f
(7.10)
6kH112rdV '
(
M
2r
or (7.10) holds for some order
2k+1 1n ) r vol (M)
2k+2 I dV a
(
-n) r vol (M)
k = 0,1,2, .
and
n-dimensional compact
Then we have
.
fM jjdAkH11
r z 1
for
M be an
Let
The equality sign of (7.9)
and
r
k
if and only if
M
is of
1.
Remark 7.1.
If
k = 0, r = p = 1,
then inequality (7.9)
is due to Reilly [2).
Proposition 7.2 also implies the following. Corollary 7.2.
For each
compact submanifold
M
Proposition 7.3. submanifold of
]Rm
(7.11)
S
.
in
Let
k =
with
IRm
M
there is no
be an
AkH = 0.
n-dimensional compact
Then we have
116kH112dV s
Xq
vol (M)
M
2k+2 (7.12)
f
IIdAk 1II2dV ` (-gn
) vol (M)
M
for
k = 0,1,2, .
holds for some
k
The equality sign of (7.11) or (7.12)
if and only if
M
is of order
We omit the proof of this proposition.
q.
§ 8. Some Applications to Spectral Geometry ¢8.
303
Some Applications to Spectral Geometry In this and the next two sections, we shall apply the
concept of order to obtain some best possible estimates of the eigenvalues of the Laplacian.
First of all, we give the following best possible estimate of
xl
of a surface up to its conformal equivalent
class.
surface which admits an order ]Rm.
M
Let
Theorem 8.1 (Chen [16)).
isometric imbedding into
1
Then, for any compact surface
conformally equivalent to
(8.1)
M
into
(8.2)
]Rm
.
IItm
is of order
be an order
Let x : M -+ iRm M
in
which is
a ll vol (M)
equality holding if and only if
imbedding of
M
we have
M,
X1 vol (M)
Proof.
be a compact Riemannian
1
1.
isometric
Then, by Theorem 6.2, we have
IHj2dV =
(4) vol (M)
M
Because the total mean curvature
f JHj2dV
is a conformal
invariant, we have
(8.3)
f_ IMI2dV = (4) vol (M) M
On the other hand, by using Reilly's inequality, we also have
6. Submanifolds of Finite Type
304
fM
(8.4)
_
a
I H12dy ? (-l) vol (M)
Thus, by combining (8.3) and (8.4), we obtain (8.1). If the equality of (8.1) holds, then the equality of Therefore, by applying Theorem 6.2,
(8.4) holds.
also of order
in
1
M
is
The converse of this follows
IItm.
immediately from Theorem 6.2 and Proposition 5.3.5.
Let
M
(Q.E.D.)
be a Clifford torus (or a square torus).
we obtain from §3.5 that
to be of order
1.
The standard
X1 vol (M) = 472.
imbedding of the Clifford torus in
M
A compact surface
is known
(m ? 4)
IItm
Then
is called
IRm
in
a conformal Clifford torus or a conformal square torus if
differs from M
Rm
by conformal mappings of
M
.
From Theorem 8.1 we obtain immediately the following Corollary 8.1 (Chen [16]).
If
M
is a conformal
Clifford torus, then we have
al vol (M) s 4v2
(8.5)
equality holding if and only if
M
admits an order
1
isometric imbedding. Let
3RPn
denote the real projective
n-space with the
standard metric. Then we have al vol (]RP2) = 127- . Veronese imbedding of IRP2 into ]R5 is an order isometric imbedding.
A compact surface
M
in
3R
TO
The 1 is
called a conformal Veronese surface if it is conformally
§ 8. Some Applications to Spectral Geometry
equivalent to the Veronese surface.
305
From Theorem 8.1 we
obtain the following (Chen (161). Corollary 8.2.
If
M
is a conformal Veronese surface,
then we have
X1 vol (M) 9 12'r ,
(8.6)
equality holding if and only if
M
admits an order
1
isometric imbedding.
From Proposition 3.5.4 and Theorem 6.2 we have the following.
Corollary 8.3.
into
IItm ,
For any isometric immersion of
we have IHIndv
r
(8.7)
IRPn
2(nn 1)
n 1
n
equality holding if and only if the immersion is of order Similarly, if we denote by
QPn
and
QPn
the complex and
n-spaces, respectively, with the
quaternion projective
standard metrics, then, by Proposition 3.5.5, 3.5.6 and Theorem 6.2, we have the following. Corollary 8.4.
into (8 .8)
IRm ,
For any isometric immersion of
we have
CPn
IH'2ndV s (2(n+1)7r)n nn n
1.
CPn
6. Submanifolds of Finite Type
306
equality holding if and only if the immersion is of order 1. Corollary 8.5.
into
IItm ,
For any isometric immersion of
we have IHI4ndV x
(8.9)
QPn
IQPn
n+
.
equality holding if and only if the immersion is of order Corollary 8.6. OP2
plane
into
1.
For any isometric immersion of the Cayley
Iltm,
we have 7
IHI16dv a (72
(8.10)
)w
OP 2
equality holding if and only if the immersion is of order
where w is defined by (4.6.49). Remark 8.1.
In Chen [16], Theorem 8.1 and Corollaries
8.1 and 8.2 are stated in a slightly different form in which the volume of
M was normalized.
1,
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces
307
Spectra of Submanifolds of Rank-one Symmetric Spaces
§9.
In this section, we will again apply the concept of order of submanifolds to obtain several best possible estimates of
xk
for
submanifolds in rank one symmetric spaces. Theorem 9.1.
M
Let
(Chen [22].)
pact submanifold of a hypersphere
be an n-dimensional comof radius
Sm(r)
r
Rm+1
in
Then
M
if
(1)
is mass-symmetric in
xp = 2 if and only if
M
Sm(r),
is minimal in
then
Sm(r)
X 1i xp 9E - 2 r
and
and hence
r
M
is of 1-type.
if
(2)
and only if Proof.
of M
in
is of finite type, then xq z 2 and Xq = 2 if
M
M
is of 1-type.
Denote by
H
and in
Itm+l
(9. 1)
H'
and Sm(r),
the mean curvature vectors respectively.
Then we have
IHI2 = IH'I2 + r-2.
Hence, by applying Theorem 6.3, we find x
(r )
(9-2)
This shows that imply
H' = 0,
X
q
z n/r2. M
that is,
a result of Takahashi, clear.
fM
vol (M)
M
Thus, Statement (2)
I H 12 dV
q
(n) vol (M) .
If xq = n/r2, then (9.1) and (9.2) is minimal in is of 1-type.
Sm.
Consequently, by
The converse of this is
is proved.
For Statement (1), we assume that the centroid of
center of S. m
M
is the
Without loss of generality, we may assume that
Sm
6. Submanifolds of Finite Type
308
From Lemma 4.5.1 and Proposition 6.1,
is centered at the origin.
we have
n (x , H) = (x , A x)
n vol (M)
(9.3)
q
Ep at llxtll2 ? ap M
Since
lies in
we find
Sm,
ilxll2.
11x112 = r2 vol (M).
Thus, by
(9.3), we obtain (9.4)
-
r2
xp.
If the equality of (9.4) holds, then the inequality of (9.3) becomes equality.
M
Thus,
is of 1-type.
The converse
of this is clear.
(Q.E.D.)
From Theorem 9.1, we obtain immediately the following.
(Chen
[221.)
M
If
Corollary 9.1.
is a compact, n-dimensional mass-
symmetric submanifold of a unit hypersphere al 9 n,
equality holding when and only when M Let
Corollary 9.2.
manifold with imbedding of q
Sm
M
Rm+l
then
is of order 1.
M be an n-dimensional compact Riemannian
for some integer
n
Xq
in
q.
into a unit hypersphere
is an imbedding of order
Then every isometric
Sm 2
Rm+l
of order
q.
Now, we need the following. Lemma 9.1.
Let
imbedding of R Pm
p
into
R Pm 4 H(m+l; R) H(m+l ; R).
be the first standard
Then an n-dimensional minimal
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces submanifold of
12 Pm
geodesic R Pn
in
M
is of 1-type if and only if If this case occurs,
IEtPm.
M
309
is a totally is of order 1
in H (m+l ; P) . Proof.
Assume that
is of 1-type in
M
H(m+l; H2).
Then
by Takahashi's result;
M
is minimal in a hypersphere
S(r)
Denote by
C
the center of
C
H (m+l ; IR) .
S (r) .
Since
of
is a
symmetric matrix, by choosing a suitable coordinates of H(m+1;JR), we may assume that
C
For any A E cp(M),
is diagonalized.
mean curvature vector of
M
H(m+l ;3R)
in
at
A
the
is given by
H = 2 (A-C) . r
(9. 5)
Because
M
is minimal in
We put
AC = CA.
Thus, we have
H E TA (Ht PI) .
IR Pm,
L = (Z E H(m+1 ;IR) ,ZC = CZ).
Then
M c L.
Let O
C1I
CiIi
C =
(9.6)
0
where A2 = A
Ii
and
are identity matrices. tr A = I.
fl L =
For each
A E cp(1
Ai = Ai and A
tr Ai = 1
r
From this, we obtain the following disjoint union:
r
cp (H2 Pm)
where
m),
O
Ai 0
I
k k
Thus, we find
Al T (IR Pm)
C
n L = U wi, i=1
we have
310
6. Submanifolds of Finite Type
O
O
Ai
Wi = O
,
O
It is clear that each
W.
R Pm
Thus, each
H(m+l ; R).
in
Ai = A1 and tr Ai = 1
I
is a totally geodesic submanifold of is a real projective space
W.
k.
R P
1
H(ki+l; R).
in
M
we see that
M
Since
lies in a
is connected and
m)
n L,
1 1 i s r.
Since
M
is totally geodesic in
R Pm,
M
for some
Wi
M c ca (R P
i,
k.
is minimal in
R Pm
and R P
1
k.
is minimal in R P 1
which lies in
lies in the hypersphere of
H(ki+l; R).
H(ki+l ;1R)
radius
r
as a minimal submanifold.
vector
H
of
for
in H (ki+l ; R)
M
M
Moreover,
centered at
c I i
with
i
Thus, the mean curvature
is given by H = -2 (A -a r
Thus, by Lemma 4.6.4, we have
A E cp(M).
2(nnl)r4 = 2 - ai - 2 (ki+1) ai. On the other hand, because
mH1 = r,
we also have
r2 = 1/1H12
Thus, we find
n/2 (n+l) .
(n+l) (ki+l) a2 + 2 (n+l) ai - 1 = O. Since
Thus, we find that R Pm.
M
a O.
is real, the discriminate of this equation is
ai
is
n 3 ki. R P
Therefore,
1.
Since
Thus,
M
M
M
lies in
this implies
is a totally geodesic
is of order 1 in
verse of this is clear.
k. I RP 1,
H(m+l ; R).
R Pn
in
The con(Q.E.D.)
311
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces For CR-submanifold of
we also have the following
CPm,
result of Ros [1). Lemma 9.2.
of
where
CPm,
L
M be a minimal, n-dimensional CR-submanifold
CPm
is imbedded in
standard imbedding.
M
Then
by its first
H(m+1; C)
is of 1-type in
if and
H(m+l ;C)
only if one of the following two cases holds: a)
M
is a totally geodesic complex submanifold of
b)
M
is a totally real submanifold of a complex n-dimen-
sional totally geodesic complex submanifold of
Then
Q
M
We suppose that
Proof.
denote the center of
S(r),
CPm.
is of 1-type in
is minimal in a hypersphere
M
S(r)
of
CPm,
H(m+l; C).
H(m+l ;C).
we can suppose that
Q
Let
is a
diagonal matrix, othersise we can use an isometry of H(m+l ;C) of the type
Since
matrix.
vector
H
A E T (M) . Thus,
A '+ PAP-1,
M
,ZQ = QZ).
of
M
with P E U(m+l),
is minimal in
in
M
S(r),
to obtain a diagonal the mean curvature
satisfies
H (m+l ; C)
H = (A - Q) /r2,
we have AQ = QA
Since H ETA (CPm),
is contained in the linear subspace We put 0
alll
Q =
a
arIr
O
Then we have the following disjoint union:
r
cp (CPm)
f1 L = U Wi, i=1
for
for
A E cp (M) .
L = (Z E H(m+l ; C)
312
6. Submanifolds of Finite Type
where o
o
E H (m+l ; C)
A.
Wi = O
I
Ai = Ai and
tr Ai =1
O
Each of these components is evidently a totally geodesic complex submanifold of
CPm (it is a
submanifold of a component
k s m)
CPk,
M
and
Consequently, the problem is
CPk.
reduced to the study of minimal CR-submanifolds of
are minimal in some hyperaphere of H(k+1;C) al,
a E ]R,
is a minimal
and whose radius is
CPk
which
whose center is
r.
From Lemma 4.9.3, we have
IHI2 = r-2 =
(9.7)
C a = dimk
n (n2+n+2a),
On the other hand, we have
H = - 2 (A - a I).
Thus, we find
r
r2 = IHI2r4 = < A - a I , A - a I > = 2 (k+l) a2 - a + 2
(9.8)
Combining (9-7) and (9.8), we find (n2+n+2a) (k+l) a2 - 2(n2+n+2a) a + (n+2a) = O. Since the discriminate of this equation must be
n2 s (n + 2a) k.
Because
implies that either 2a = n,
n2
that is,
(n + 2a) k,
And if M
that is,
M
n = 2k.
This
is totally real or
is a complex submanifold of
we find that if M
is complex,
we get
k a n - a, we get a (2a - n) ? O.
a = 0,
M
? 0,
CPk.
Because
is totally real, then
If the first case occurs,
k = n.
M
is
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces
a totally real submanifold of a totally geodesic M
If the second case occurs,
in
aPn
is a totally geodesic
313
CP
QPm
k
with
n = 2k.
Conversely, if
then
M
M
is a totally geodesic
is of order 1 in
any orthonormal basis
C) c H (m+l ; a) .
H (k+l
totally real submanifold of
aPk
aPn,
in
CPm
If
M
is a
then for any A E T(M)
E1,...,En
of
TA M,
form an orthonormal basis of TA (aPn).
and
E1....,En, J El,...,J En
Therefore, by Theorem
4.6.1 and (4.6.26), we obtain H = 2(1 - (n+l) A)/n. This implies that
M
is of 1-type in
H (n+l ; C) C H (m+l ; C) .
(Q. E. D. )
In the following, we give some best possible estimate of
al
for compact minimal submanifolds of projective spaces. Theorem 9.2.
(Chen [24].)
minimal submanifold of tional curvature 1.
M
Laplacian of
R Pm,
M be a compact, n-dimensional,
Let
where R Pm
is of constant sec-
Then the first non-zero eigenvalue
11
of the
satisfies
%1 s 2 (n+l) ,
(9.9)
equality holding if and only if
M
is a totally geodesic
it Pn
in
IIt Pm
Proof.
Let
fold of P Pm.
M
be a compact, n-dimensional, minimal submani-
Then, by Lemma 4.6.5, we have
IH12 = 2(n+l)/n.
Thus, by Theorem 6.2, we obtain (9.9). If the equality of (9.9)
holds, then Theorem 6.2 implies that M
is of 1-type.
Thus, by
6. Submanifolds of Finite Type
314
applying Lemma 9.1, we conclude that in
IIiPn
is a totally geodesic
(Q.E.D.)
The converse of this is clear.
R Pm.
Theorem 9.3. (n a 2),
M
(Chen [24].)
M be an n-dimensional
Let
compact, minimal submanifold of
CPm,
of constant holomorphic sectional curvature 4.
(9.10)
QPm
where
Then we have
%1 s 2(n+2),
equality holding if and only if (1)
n
is even,
M
(2)
is a
n
CP2
and (3)
M
is a complex totally geodesic submanifold of
CPm
Proof. Let
QPm
be isometrically imbedded in
its first standard imbedding.
minimal submanifold of
CPm,
(9.11)
IHI 2
by
is a compact, n-dimensional,
then, by Lemma 4.6.5, we obtain 2 n+2 n
equality holding if and only if submanifold of
M
If
H(m+l ;C)
n
is even and
M
is a complex
By combining (9.11) with inequality (6.7)
LPm.
of Reilly, we obtain (9.10).
If the equality sign of (9.10) holds, then the equality sign of (9.11) holds.
submanifold of
n
is even and
M
is a complex
On the other hand, we also have
CPm. p
(9.12)
Thus,
J
k
dV = (n) vol (M).
M
Thus, by applying Theorem 6.2, we conclude that
M
is of order 1
§ 9. Spectra of Submanifolds of Rank-one Symmetric Spaces in
315
Therefore, by applying Lemma 9.2, we conclude
H(m+l ;C).
n
M
that
is a
which is imbedded in
CP2
QPm
Conversely, if
geodesic complex submanifold.
as a totally
M
is a
n
then we have
CP2,
Remark 9.1.
Yang and Yau [1] showed that if
holomorphic curve in
CPm,
holding if and only if
M
M
is a
Moreover, Ejiri [2)
is a compact, n-dimensional
M
Kaehler submanifold of
then
a1 a 2(n+2),
equality
is a totally geodesic complex sub-
CPm.
Theorem 9.4. sional
a1 1 8.
then
CPm,
and Ros [2] proved that if
manifold of
(Q. E. D.)
X1 = 2(n+2).
(n ? 4),
(Chen [24].)
Let
M be a compact, n-dimen-
minimal submanifold of
QPm,
QPm
is of
Then we have
constant guaternion sectional curvature 4.
(9.13)
where
a1 s 2(n+4),
equality holding if and only if (1) n
is a multiple of 4,
(2)
M
n
j
QP4,
AD-d (3)
M
is imbedded in
QPm
as a totally geodesic
guaternionic submanifold. Proof.
manifold of
(9.14)
Let QPm.
M be a compact, n-dimensional, minimal subThen, Lemma 4.6.6 implies IHI2 s 2(n+4)
n
Therefore, by combining (9.14) with Theorem 6.2, we obtain (9.13).
6. Submanifolds of Finite Type
316
Now, if the equality sign of (9.13) holds, then (9.14) becomes equality.
Thus, by Lemma 4.6.5,
is a quaternionic submanifold of
n
is a multiple of 4 and M Thus, by a result of
QPm.
n
Gray [1], we conclude that
M
is a totally geodesic
The converse of this is clear.
QPm.
Remark 9.2.
QP4
in
(Q.E.D.)
Recently, Martinez, Perez, and Santos informed
the author that they can also obtain (9.13) for compact, generic,
minimal submanifolds of
QPm.
Similarly, by using (4.6.46) and Theorem 6.2, we may also obtain the following. Theorem 9.5.
(Chen [24].)
Let
M
be a compact, n-dimen-
sional, minimal submanifold of the Cavlev Plane is of maximal sectional curvature 4.
OP2, where
OP2
Then we have
x1 s 4n.
(9.15)
For CR-submanifolds, we also have the following Proposition 9.1.
(Ejiri [2] and Ros [1].)
Let M be a
compact, n-dimensional, minimal. CR-submanifold of
CPm.
Then
we have
(9.16) where
x1 a
s 2 (n2 + n + 2a) /n,
is the complex dimension of the holomorphic distribution.
This Proposition follows easily from Lemma 4.9.3 and Theorem 6.2.
Similarly, by using 4.9.4 and Theorem 6.2, we have the
following.
§ 9. Spectra of Submanifolds of Rank-one Symmetric Spaces Proposition 9.2.
M be a compact, n-dimensional,
Let
minimal CR-submanifold of
QPm.
Then we have
ll s 2(n2+n+12a)/n,
(9.17) where
317
a
is the guaternionic dimension of the guaternion dis-
tribution. Theorem 9.6.
of
IR Pm,
where
M
Let
is imbedded in H (m+l ; It)
I2 Pm
standard imbedding.
be an n-dimensional, compact submanifold
M
If
is of finite type in
by its first H(m+l ; I2),
then
we have
(9.18)
X
a 2(n+l),
q
equality holding if and only if in
I2Pm.
Proof.
If this case occurs,
M
is a totally geodesic P Pn
q = p = I.
From Theorem 6.3, we have
(9.19)
p J
H j
2
d
q
I
3 (n) vol (M).
M
Moreover, from Lemma 4.6.4, we also have IHI2 a 2(n+l)
(9.20)
n
Combining (9.19) and (9.20), we obtain (9.18). If the equality of (9.18) holds, then (9.19) and (9.20) become equalities.
Thus, by Theorem 6.3 and Lemma 4.6.4, we see
that M is minimal in
It Pm
and M is of 1-type in
Thus, by Lemma 9.1, we obtain the theorem.
H (m+l ; ]R) . (Q.E.D.)
6. Submanifolds of Finite Type
318
of
CPm,
where
dard imbedding.
M be an n-dimensional, compact submanifold
L
Theorem 9.7.
is imbedded in
CPm
M
If
H(m+l ; C)
by its first stan-
is of finite type, then we have
Xq x 2 (n+l) .
(9.21)
The equality of (9.21) holds if and only if
M
is a minimal to-
tally real submanifold of a totally geodesic complex submanifold CPn
of
CPm.
Proof.
Then, by Theorem 6.3, we see that the mean curvature vector
CPm. H
Let M be an n-dimensional, compact submanifold of
of
M
in
H (m+l
satisfies
C)
a
(9.22)
J
H 12 dv
n ) vol (M) .
M
Thus, by combining (9.22) with Lemma 4.6.4, we find
(9.23)
Xq z 2(n+l),
equality holding if and only if in
CPm
and M
is of 1-type in
M
is totally real and minimal
H(m+l ;C).
Thus, by using
Lemma 9.2, we obtain the theorem.
(Q.E.D.)
Similarly, we also have the following.
of
QPm,
where
QPm
standard imbedding.
(9.24)
M be a compact. n-dimensional submanifold
L
Theorem 9.8.
is imbedded in If
M
H (m+l ; Q)
by its first
is of finite type, then we have
Xq 6 2 (n+l) ,
99. Spectra of Submanifolds of Rank-one Symmetric Spaces
equality holding if and only if
M
submanifold of a totally geodesic
319
is a minimal totally real QPn
in
QPm
6. Subnwnifolds of Finite Type
320 §10.
Mass-symmetric Submanifolds From Theorem 9.1 and Corollary 9.1, we have a best possible
estimate of
for mass-symmetric submanifolds of a hypersphere.
X 1
In this section, we shall study
Xp
for mass-symmetric submani-
folds in projective spaces. Theorem 10.1.
be isometrically imbedded in
L g t R Pm
by its first standard imbedding.
H(m+l ;]R)
If M
n-dimensional, mass-symmetric submanifold of R Pm,
is a compact, then
2n m+1
(10.1)
m
1
equality holding if and only if
Since R Pm
Proof.
n = m
and
M = R Pn
is isometrically imbedded in
H (m+l ; R)
b y its first standard imbedding, Theorem 4.6.1 implies that
R Pm
is imbedded as a minimal submanifold in a hypersphere
of
radius
r = [m/2(m+1)]1/2. R Pm
centroid of
Thus, by Lemma 4.3, we see that the
is the center of
M
the centroid of
S(r)
S(r).
Thus, by the hypothesis,
is the center of the hypersphere
There-
S(r).
fore, by applying Theorem 9.1, we obtain the inequality (10.1). If the equality sign of (10.1) holds, then, by Theorem 9.1,
M
is of 1-type in
M
Therefore,
we obtain
and
M
is also minimal in R Pm.
we conclude that x1 = 2(n+l).
H(m+l ; R)
M
and
M = R Pn.
S(r).
By applying Lemma 9.1,
is a totally geodesic R Pn
On the other hand, we have
n = m
is minimal in
in R Pm. Hence
al = 2n(m+l)/m.
Thus,
The converse of this is clear. (Q. E. D. )
§ 10.
M
symmetric submanifold of
be an n-dimensional. compact, massIF Pm,
isometrically imbedded in imbedding.
H(m+l;IF)
s ap
1
%p = 2n(m+l)/m
IF Pm
is
by its first standard
2n m+1 m
if and only if
minimal totally real submanifold of
to be of order Proof.
where
Then we have
(10.2) Moreover,
= Q or Q,
IF
[p , q]
Since
in
IF Pm
m = n
and
Where
IF Pn.
M
M
is a
is assumed
H (m+l ; IF) . is isometrically imbedded in
H(m+l; IF)
by its first standard imbedding, Theorem 4.6.1 implies that
IF Pm
is imbedded as a minimal submanifold in a hypersphere
of
r = rm/2(m+1)]1/2.
radius IF Pm
is the centroid of
S(r)
Thus, by Lemma 4.3, the centroid of
in H (m+l ; IF) .
S (r)
hypothesis, the centroid of
M
Hence, by the
is also the centroid of
S(r).
Therefore, by applying Theorem 9.1, we obtain inequality (10.2). if
%p = 2n(m+l)/m,
minimal in M
then Theorem 9.1 implies that
S (r) and hence
is also minimal in
is of 1-type in
M
IF Pm.
IFPm
manifold of a totally geodesic case occurs, then to (10.2). find
n = m
X
l
is
H (m+l ; IF) .
Thus,
By applying Lemma 9.2 and its quater-
nionic version, we conclude that invariant submanifold of
M
M
is either a totally geodesic
or a totally real minimal subIF Pn
in
IF Pm.
= 2(n+d), d = 2 or 4.
If the second case ocuurs, by our assumption.
If the first
This contradicts
ap = 2(n+l).
Thus, we
6. Submanifolds of Finite Type
322
M
Conversely, if
is a totally real minimal submanifold of
then, by (4.6.26) and the fact that
]F Pn,
submanifold of the hypersphere [n/2(n+1)]1/2,
vector
H
IF Pn
-IL = 2 (n+l)
.
r
with radius
S(r)
is a minimal r =
we may conclude that M has mean curvature H = H,
satisfying
vector of
iF Pn
where
ft
is the mean curvature
Thus, we obtain
in H (n+l ; iF) .
ap = (Q.E.D.)
Similarly, by using Remark 4.6.2 and Theorem 9.1, we have the following.
Let
Theorem 10.3.
M be a compact, n-dimensional
mass-symmetric submanifold of ly imbedded in
H(3 ;Cay)
OP2,
Where
OP2
(n z 2),
is isometrical-
by its first standard imbedding.
Then we have xl a 3n,
(10.3)
equality holding if and only if surface of
OP2.
M
is a minimal, totally real
Here, by a totally real surface of
OP2,
A&
mean a surface whose tangent planes are totally real with respect to the Cavley structure of
OP2.
Theorems 10.1, 10.2, 10.3 together with corollary 9.1 give the best possible upper bound of
11
for compact mass-symmetric
submanifolds in rank-one symmetric spaces. From Theorem 10.1, we have the following. Corollary 10.1.
R Pn
cannot be isometrically imbedded in a
Submanifolds
§ 10.
as a mass-symmetric submanifold for
IR Pm
Proof.
m' n.
can be isometrically imbedded in
IR Pm
Et Pm
as a mass-symmetric submanifold, then Theorem 10.1 implies
m >n,
l
If
323
2n(m+l)/m.
This contradicts to the fact that
X1 = 2(n+l).
(Q.E.D.) Although,
P n
can be isometrically imbedded in
as a mass-symmetric submanifold in a natural way,
be isometrically imbedded in submanifold.
cannot
as a mass-symmetric
This result is a special case of the following.
Corollary 10.2.
M
L
Riemannian manifold with
be a compact, n-dimensional,
>l ? 2(n+l).
isometrically imbedded in unless
IF Pm, m >n,
P Pn
]F P n
m = n, al = 2(n+l)
real minimal submanifold in
IF Pm
and
Then
M
cannot be
as a mass-symmetric submanifold M
can be imbedded as a totally
IF Pn.
This Corollary follows immediately from Theorem 10.2.
BIBLIOGRAPHY
Abe, K. [1]
Applications of Riccati type differential equation to Riemannian manifolds with totally geodesic distribution, Tohoku Math J., 25 (1973), 425-444.
Adem, J. [1]
Relations on iterated reduced powers, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 636-638.
Asperti, A.C. [1]
Immersions of surfaces into 4-dimensional spaces with nonzero normal curvature, Ann. Mat. Pura Appl. 125 (1980), 313-328.
Asperti, A.C., Ferus, D., and Rodriguez, L. [1]
Surfaces with non-zero normal curvature tensor, to appear.
Atiyah, M.F., and Lawson, H.B. [1]
In preparation.
Atiyah, M.F., and Singer, I.M. [11
The index of elliptic operators, III, Ann. of Math., 87 (1968), 546-604.
Banchoff, T.F. [1]
The two-piece property and tight n-manifolds-with boundary in Em, Trans. Amer. Math. Soc. 161 (1971), 259-267.
Barros, M., Chen, B.-Y., and Urbano, F. [1]
Quaternion CR-submanifolds of quaternion manifolds, Kodai
Math. J., 4 (1981), 399-417.
Barros, M. and Urbano, F. [11
Submanifolds of complex Euclidean space which admit a holomorphic distribution, Quart. J. Math. Oxford, (2) 34 (1983), 141-143.
[2]
Topology of quaternion CR-submanifolds, Boll. Un. Mat. Ital., (6) 2-A (1983), 103-110.
326
Bibliography
Bejancu, A. [1]
CR-submanifolds of a Kaehler manifold, I, Proc. Amer. Math. Soc., 69 (1978), 135-142.
[2]
CR-submanifolds of a Kaehler manifold, II, Trans. Amer. Math. Soc, 250 (1979), 333-345.
[3)
On the.geometry of leaves on a CR-submanifold, An. St. Univ. "Al. I. Cuza" Iasi, 25 (1979), 393-398.
Berger, M. [1]
Geometry of spectrum, I, Proc. Symp. Pure Math. Vol. 27, Part 2,
[2]
(1975), 129-152.
Sur lea premieres valeurs propres des varietes riemanniennes, Compositio Math., 26 (1973), 129-149.
Berger, M., Gauduchon, P., and Mazet, E. [1]
Le Spectre d'une Variete Riemannienne, Math. 194 (1971), Springer, Berlin.
Lecture Notes in
Besse, A.L. [1]
Manifolds all of whose Geodesics are Closed, Ergeb. Math. Grenz. 93 (1978), Springer, Berlin.
Blair, D.E., and Chen, B.-Y. [1]
On CR-submanifolds of Hermitian manifolds, Israel J. Math., 34 (1979), 353-363.
Blaschke, W. [1]
Vorlesungen fiber Differentialgeometrie III, Springer, 1929, Berlin.
Sleeker, D., and Weiner, J. [1]
Extrinsic bounds on X1 of A on a compact manifold, Comm. Math. Helv., 51 (1976), 601-609.
Borsuk, K. [1]
Sur la courbure totale des courbes, Ann. Soc. Math. Polon., 20 (1947), 251-256.
Calabi, E. [1]
Isometric imbedding of complex manifolds, Ann. of Math., 58 (1953), 1-23.
Bibliography
327
Cartan, 9. [l]
Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France, 54 (1926), 214-264 and 55 (1927), 114-134.
[2]
Lecons sur la Geometrie des Espaces de Riemann, GauthierVillars, Paris, 1928.
Carter, S. and West, A. [1]
Tight and taut immersions, Proc. London Math. Soc., 25 (1972), 701-720.
Cecil, T.E. and Ryan, P.J. [1]
Tight sperical embedding, Lecture Notes in Math., 838 (1981), 94-104.
Chen, B.-Y. [1)
On an inequality of T.J. Willmore, Proc. Amer. Math. Soc., 26 (1970), 473-479.
[2)
On total curvature of immersed manifolds, I, Amer. J. Math., 93 (1971), 148-162.
[3)
On a variational problem of hypersurfaces, J. London Math. Soc., 6 (1972), 321-325.
[4)
G-total curvature of immersed manifolds, J. Differential Geometry, 7 (1972), 373-393.
[5]
On the total curvature of immersed manifolds, II, Amer. J. Math., 94 (1972), 899-907.
[6]
On an inequality of mean curvature, J. London Math. Soc., 4 (1972), 647-650.
[7]
Geometry of Submanifolds, Dekker, 1973, New York.
[8]
An invariant of conformal mappings, Proc. Amer. Math. Soc., 40 (1973), 563-564.
[9]
On the total curvature of immersed manifolds, III, Amer. J. Math., 95 (1973), 636-642.
[10]
Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital., (4) 10 (1974), 380-385.
[11]
Mean curvature vector of a submanifold, Proc. Symp. Pure Math. Vol. 27, Part I, (1975), 119-123.
Bibliography
328 [12]
Some relations between differential geometric invariants and topological invariants of submanifolds, Nagoya Math. J., 60 (1976), 1-6.
[13]
Total mean curvature of immersed surfaces in Amer. Math. Soc., 218 (1976), 333-341.
[14]
Totally umbilical submanifolds of quaternion-space-forms, J. Austra. Math. Soc. (Series A), 26 (1978), 154-162.
[15]
On total curvature of immersed manifolds, IV, Bull. Inst. Math. Acad. Sinica, 7 (1979), 301-311.
[16]
Conformal mappings and first eigenvalue of Laplacian on surfaces, Bull. Inst. Math. Acad. Sinica, 7 (1979), 395-
Em, Trans.
400. [17]
Geometry of Submanifolds and Its Applications, Sci. Univ. Tokyo, 1981, Tokyo.
[18]
Differential geometry of real submanifolds in a Kaehler manifold, Monatsch. fiir Math., 91 (1981), 257-274.
[19]
On the total curvature of immersed manifolds. V. Bull. Inst. Math. Acad. Sinica, 9 (1981), 509-516.
[20]
CR-submanifolds of a Kaehler manifold, I, II, J. Differential Geometry, 16 (1981). 305-322 and 493-509.
[21]
Cohomology of CR-submanifolds, Ann. Fac. Sc. Toulouse Math. (Series V) 3 (1981), 167-172.
[22]
On total curvature of immersed manifolds, VI, Bull. Inst. Math. Acad. Sinica, 11 (1983), 309-328.
[23]
Review of "Total Curvature in Riemannian Geometry" by Willmore, Bull. London Math. Soc. 16 (1984).
[24]
On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one symmetric spaces, Chinese J. Math., 11 (1983).
[25]
On submanifolds of finite type, Soochow J. Math., 9 (1983).
[26]
Some non-integrability theorems of holomorphic distributions, Alg. Diff. Topology-Global Diff. Geom. Appl., Teubner-Verlag, 1984, Leipzig.
Chen, B.-Y. and Houh, C.S. [1]
On stable submanifolds with parallel mean curvature, Quart.
J. Math. Oxford,
(3)
26 (1975), 229-236.
Bibliography [2]
329
Totally real submanifolds of a quaternion projective space, Ann. Mat. Pura Appl., 120 (1979), 185-199.
Chen, B.-Y., Houh, C.S., and Lue, H.S. [1]
Totally real submanifolds, J. Differential Geometry 12 (1977) ,
473-480.
Chen, B.-Y., Ludden, G.D., and Montiel, S. [1]
Real submanifolds of a Kaehler manifold, (preprint).
Chen, B.-Y. and Nagano, T. [1]
Totally geodesic submanifolds of symmetric spaces, I, II, Duke Math. J., 44 (1977), 745-755; 45 (1978). 405-425.
[2]
Harmonic metrics, harmonic tensors and Gauss maps, J. Math. Soc. Japan, 36 (1984).
Chen, B.-Y. and Ogiue, K. (1]
On totally real submanifolds, Trans. Amer. Math. Soc., 193 (1974), 257-266.
[2]
Two theorems on Kaehler manifolds, Michigan Math. J., 21 (1974), 225-229.
Chen, B.-Y. and Vanhecke, L. [1]
The spectrum of the Laplacian of Kahler manifolds, Proc. Amer. Math. Soc., 79 (1980), 82-86.
[2]
Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67.
Chen, B.-Y. and Verheyen, P. [1]
Totally umbilical submanifolds of Kaehler manifolds, II, Bull. Soc. Math. Belg. (to appear).
[2]
Sous-varietes dont les sections normales sont des geodesiques, C.R. Acad. Sc. Paris, 293 (1981). 611-613.
Chen, B.-Y. and Yano, K. [1]
On the theory of normal variations, J. Differential Geometry, 13
(1978), 1-10.
Cheng, S.Y. [1]
Eigenfunctions and eigenvalues of Laplacian, Proc. Symp. Pure Math. Vol. 27, Part 2, (1975), 185-194.
330
Bibliography
Chern, S.S. (1)
A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math., 45 (1944), 747-752.
[2)
Differentiable Manifolds, University of Chicago, 1959.
[3)
Minimal Submanifolds in a Riemannian Manifold, Univ. Kansas, Tech. Rep. No. 19 (1968).
[4]
Complex Manifolds without Potential Theory, Springer, 1979, Berlin.
[5)
Selected Papers, Springer, 1978, Berlin.
Chern, S.S., doCarmo, M., and Kobayashi, S. [1]
Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer (1970), 59-75.
Chern, S.S. and Hsiung, C.C. [1)
On the isometry of compact submanifolds in Euclidean space, Math. Ann. 149 (1962/63), 278-285.
Chern, S.S. and Kuiper, N. [1)
Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space, Ann. of Math., 56 (1952), 422-430.
Chern, S.S. and Lashof, R.K. [1]
On the total curvature of immersed manifolds, Amer. J. Math., 79 (1957), 306-318.
[2]
On the total curvature of immersed manifolds, II,
Michigan Math. J., 5 (1958), 5-12.
Dajczer, M. [1]
Doctoral Thesis, IMPA, 1980.
doCarmo, M.P. and Wallach, N.R. [1]
Minimal immersions of spheres into spheres, Ann. of Math., 93
(1971), 43-62.
Eells, J., Jr. and Lemaire, L. [1)
A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.
Bibliography
331
Eells, J. Jr. and Sampson, H. [1]
Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.
Ejiri, N. [1)
A counter example for Weiner's open question, Indiana Univ. Math. J., 31 (1982), 209-211.
[2]
The first eigenvalue of 6 for compact minimal submanifolds in a complex projective space, to appear.
[3]
Totally real minimal immersions of n-dimensional real space form into n-dimensional complex space forms, Proc. Amer. Math. Soc., 84 (1982), 243-246.
Erbacher, J. [1]
Reduction of the codimension of an isometric immersion, J. Differential Geometry, 5 (1971), 333-340.
Escobales, R.H. [1)
Riemannian submersions with totally geodesic fibers, J. Differential Geometry, 10 (1975), 253-276.
Fary, I. [1]
Sur la courbure totale d'une courbe gauche faisant un noeud, Bull. Soc. Math. France, 77 (1949), 128-138.
Fenchel, W. [1]
Uber die Krilmmung and Windung geschlossenen Raumkurven, Math. Ann., 101 (1929), 238-252.
Ferus, D. [1]
Totale Absolutkriimmung in Differentialgeometrie and Topologie, Lecture Notes in Math., 66 (1968), springer, Berlin.
[2]
Symmetric submanifolds of Euclidean space, Math. Ann., 247 (1980), 81-93.
Gheysens, L., Verheyen, P., and Verstraelen, L. [1)
Sur les surfaces Q ou les surfaces de Chen, C.R. Acad. Sc. Paris, 292 (1981), 913-916.
[2]
Characterization and examples of Chen submanifolds, J. Geometry, 20 (1983), 47-62.
Bibliography
332
Goldberg, S.I. [1]
Curvature and Homology, Academic Press, 1962, New York.
Gray, A. [1]
A note on manifolds whose holonomy group is a subgroup Sp(1), Michigan Math. J., 16 (1969), 125of Sp(n) 128.
Guadalupe, I.V. and Rodriguez, L. [1]
Normal curvature of surfaces in space forms, Pacific J. Math., 106 (1983), 95-103.
Haantjes, J. [1]
Conformal representations of an n-dimensional euclidean space with a non-definite fundamental form on itself, Proc. Kon. Ned. Akad. Amsterdam, 40 (1937), 700-705.
Heintze, E. and Karcher, H. [1]
A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sc. Ecole Norm. Sup., 11 (1978), 451-470.
Helgason, S. [1]
Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978, New York.
Hersch, J. [1]
Quatre proprietes isoperimetriques de membranes spheriques homogenes, C.R. Acad. Sc. Paris, 270 (1970), 1645-1648.
Husemdller, D. [1]
Fibre Bundles, McGraw-Hill, 1966, New York.
Houh, C.S. [1]
On spherical
c7-submanifolds,
Chinese J. Math., 2 (1974),
128-134.
Hsiung, C.C. [1]
Some integral formulas for hypersurfaces, Math. Scand. 2 (1954), 286-294.
[2]
Isoperimetric inequalities for 2-dimensional Riemannian manifolds with boundary, Ann. of Math., 73 (1961), 213-220.
Bibliography
333
Klingenberg, W. [1]
Lectures on Closed Geodesics, Springer, 1978, Berlin.
[2]
Riemannian Geometry, Walter de Gruyter, 1982, Berlin.
Kobayashi, S. [1]
Imbeddings of homogeneous spaces with minimum total curvature, Tohoku Math. J. 19 (1967), 63-70.
[2]
Isometric imbeddings of compact symmetric spaces, Tohoku Math. J. 20 (1968), 21-25.
[31
Transformation Groups in Differential Geometry, Springer, 1972, Berlin.
[4]
Recent results in complex differential geometry, Jber. Deut. Math. Verein, 83 (1981), 147-158.
Kobayashi, S. and Nomizu, K. [1]
Foundations in Differential Geometry, I, II, WileyInterscience, 1963 and 1969, New York.
Kuiper, N.H. [1]
Immersions with minimal total absolute curvature, Colloq. Geom. Diff. Globale (Bruxelles, 1958), 75-88.
[2]
Tight imbeddings and maps, submanifolds of geometrical class three in EN, Chern Symposium, (1979), 97-146, Springer, Berlin.
Kuiper, N.H. and Meeks, W.H. [1]
Total absolute curvature for knotted surfaces, to appear.
KUhnel, W. [1]
Die totale mittlere Kr6mmung gewisser Hyperflachen im Monatsch. fur Math., 87 (1979), 123-131. 3R 5,
[2]
A lower bound for the ith total absolute curvature of an immersion, Colloq. Math., 41 (1979), 253-255.
Langevin, R. and Rosenberg, H. [1]
On curvature integrals and knots, Topology, 15 (1976), 405-416.
Lashof, R.K. and Smale, S. [1]
On the immersions of manifolds in Euclidean spaces, Ann. of Math., 68 (1958), 562-583.
334
Bibliography
Lawson, Jr., H.B. [1]
Rigidity theorems in rank-1 symmetric spaces, J. Differential Geometry, 4 (1970), 349-357.
[2]
Complete minimal surfaces in
S3,
Ann. of Math.,
(1970). 335-374.
Li, P. and Yau, S.T. [1]
A new cohformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269-291.
Lichnerowicz, A. (1)
Equations de Laplace et espaces harmoniques, Premier Colloque Equations Derives Partielles, Louvain, 1953, 9-23.
Little, J. [1)
On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl., 88 (1969). 261336.
[2]
Manifolds with planar geodesics, J. Differential Geometry, 11 (1976). 265-285.
Little, J. and Pohl, W.F. [1]
On tight immersion of maximal codimension, Invent. Math., 13 (1971), 179-204.
Maeda, M. [1]
The integral of the mean curvature, Sci. Rep. Yokohama Nat. Univ., Sect. I. 25 (1978), 17-21.
Masal'cev, L.A. [1]
The first eigenvalue of the Laplacian of a compact orientable hypersurface of a Euclidean space, Ukrain. Geom. Sb. Vyp., 19 (1976). 104-108.
McKean, H.P. and Singer. I.M. [1]
Curvature and the eigenvalues of the Laplacian, J. Differential Geometry, 1 (1967). 43-69.
Milnor, J.W. [1]
On the total curvature of knots, Ann. of Math., 52 (1950). 248-257.
Bibliography
335
[2]
On manifolds homeomorphic to the 7-sphere, Ann. of Math., 69 (1956), 399-405.
[3]
Morse Theory, Ann. Math. Studies, 51 (1963), Princeton Univ. Press, Princeton.
[4]
Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542.
Moore, J.D. [1]
Isometric immersions of Riemannian products, J. Differential Geometry, 5 (1971), 159-168.
Morse, M. [1]
The Calculus of Variations in the Large, Amer. Math. Soc., 1934, New York.
Morvan, J.-M. [1]
Topology of a submanifold and external curvatures, Rend. Mat. 13 (1980), 1-20.
Mostow, G.D. [1]
Equivariant embeddings of Euclidean space, Ann. of Math., 65 (1957), 432-446.
Nagano, T. [1]
On fibered Riemannian manifolds, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 10 (1960), 17-27.
[2]
On the minimal eigenvalues of the Laplacians in Riemannian manifolds, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 11 (1961), 177-182.
[3]
Homotopy invariants in differential geometry, memoir Amer. Math. Soc., 100 (1970).
Naitoh, H. and Takeuchi, M. [1]
Totally real submanifolds and symmetric bounded domains, Osaka J. Math., 19 (1982), 717-731.
Nakagawa, H. [1]
On a certain minimal immersion of a Riemannian manifold into a sphere, Kodai Math. J., 3 (1980), 321-340.
Nash, J.F. [1]
C1-isometric imbeddings, Ann. of Math., 60 (1954), 383-396.
Bibliography
336
Nomizu, K. and Yano, K. [1]
On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163-170.
Obata, M. [i]
Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14 (1962), 333340.
Ogiue, K. [1]
Differential geometry of Kaehler submanifolds, Adv. in Math., 13 (1974), 73-114.
[2]
Positively curved totally real minimal submanifolds immersed in a complex projective space, Proc. Amer. Math. Soc., 56 (1976), 264-266.
O'Neill, B. [1]
The fundamental equations for a submersion, Michigan Math. J., 13 (1966), 459-469.
Osserman, R. [1]
A Survey of Minimal Surfaces, Van Nostrand, 1969, Princeton.
Otsuki, T. [1]
On the existence of solutions of a system of quadratic equations and its geometrical applications, Proc. Japan Acad., 29 (1953), 99-100.
[2]
On the total curvature of surfaces in Euclidean spaces, Japan, J. Math., 35 (1966), 61-71.
[3]
Surfaces in the 4-dimensional Euclidean space isometric to a sphere, Kodai Math. Sem. Rep., 18 (1966), 101-115.
Palais, R.S. [1]
Imbeddings of compact differentiable transformation groups in orthogonal representations, J. Math. Mech., 6 (1957),
673-678. [2]
Seminar on the Atiyah-Singer Index Theorem, Ann. Math. Studies, 57 (1965), Princeton.
Patodi, V.K. [1]
Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc., 34 (1970), 269-285.
337
Bibliography
Reckziegel, H. [1]
On the problem whether the image of a given differentiable map into a Riemannian manifold is contained in a submanifold with parallel second fundamental form, J. Reine Angew. Math., 325 (1981), 87-104.
Reilly, R.C. [1]
Variational properties of mean curvatures, Proc. 13th Biennial Sem. Canad. Math. Congress, Vol. II, (1973), 102-114.
[2]
On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comm. Math. Helv., 52 (1977), 525-533.
Ros, A. [1]
Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J., 6 (1983), 88-99.
[2]
On spectral geometry of Kaehler submanifolds, to appear.
[3]
Eigenvalue inequalities for minimal submanifolds and P-manifolds, to appear.
Rouxel, B. [1]
Q-submanifolds in Euclidean space, Kodai Math. J., 4 (1981), 181-188.
[2]
Ruled Q-surfaces in Euclidean space Math., 6 (1980), 117-121.
En,
Soochow J.
Sakai, T. [1]
On eigenvalues of Laplacian and curvature of Riemannian manifolds, Tohoku Math. J., 23 (1971), 589-603.
Sakamoto, K. [1)
Planar geodesic immersions, Tohoku Math. J., 29 (1977), 25-56.
Shiohama, K. and Takagi, R. [1]
A characterization of a standard torus in ential Geometry, 4 (1970), 477-485.
E3,
J. Differ-
Simons, J. [1]
Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
Bibliography
338
Smale, S. [1)
The classification of immersions of spheres in Euclidean spaces, Ann. of Math., 69 (1959), 327-344.
Spivak, M. [1)
A Comprehensive Introduction to Differential Geometry, Vol. I-V, Publish or Perish, 1970, Berkeley.
Sternberg, S. [1)
Lectures on Differential Geometry, Prentice-Hall, 1964, New Jersey.
Sunday, D. [1)
The total curvature of knotted spheres, Bull. Amer. Math. Soc., 82 (1976), 140-142.
Tai, S.S. (1)
Minimum imbeddings of compact symmetric spaces of rank one, J. Differential Geometry, 2 (1968), 55-66.
Takahashi, T. [1)
Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380-385.
Takeuchi, M. and Kobayashi, S. [1)
Minimal imbeddings of R-spaces, J. Differential Geometry, 2 (1968), 203-215.
Tanno, S. [1)
Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J., 25 (1973), 391-403.
Thomsen, G. [1]
Ober konforme Geometrie I: Grundlagen der konformen Flichentheorie, Abh. Math. Sem. Univ. Hamburg, (1923),
31-56. Wallach, N.R. [1)
Minimal immersions of symmetric spaces into spheres, Symmetric Spaces, 1972, Dekker, 1-40.
Weiner, J.L. [1)
On a problem of Chen, Willmore, et al., Indiana Univ. Math. J., 27 (1978), 19-35.
339
Bibliography
White, J.H.
A global invariant of conformal mappings in space, Proc. Amer. Math. Soc., 38 (1973), 162-164.
[1)
Willmore, T.J. [1]
Note on embedded surfaces, An. St. Univ. Iasi, s.I.a. Mat., 11B (1965), 493-496.
[2]
Mean curvature of immersed surfaces, An. St. Univ. Iasi, s.I.a. Mat. 14 (1968), 99-103.
[3)
Mean curvature of Riemannian immersions, J. London Math. Soc., 3 (1971), 307-310.
[4)
Tight immersions and total absolute curvature, Bull. London Math. Soc., 3 (1971), 129-151.
[5]
Mean curvature of immersed manifolds, Topics in Differential Geometry, Academic Press, 1976, 149-156.
[6)
Total Curvature in Riemannian Geometry, John Wiley and Sons, 1982, Chichester.
Willmore, T.J. and Jhaveri, C. An extension of a result of Bang-Yen Chen, Quart. J. Math. Oxford, 23 (1972), 319-323.
[1]
Wintgen, P. [1]
On the total curvature of surfaces 39 (1978), 289-296.
[2]
Sur l'inequalite de Chen-Willmore, C.R. Acad. Sc. Paris, 288 (1979), 993-995.
in E4,
Colloq. Math.,
Witt, E. [1]
Eine Identitat zwischen Modulformen zweiten Grades, Abh. Math. Sem. Univ. Hamburg, 14 (1941), 323-337.
Wolf, J.A. [1)
Elliptic spaces in Grassmann manifolds, Illinois J. Math., 7 (1963), 447-462.
[2)
Spaces of Constant Curvature, McGraw-Hill, 1967, New York.
Yamaguchi, S. [1)
Remarks on the scalar curvature of immersed manifolds, Kodai Math. Sem. Rep., 26 (1975), 240-244.
Bibliography
340
Yang, P.C. and Yau, S.T. [1]
Eigenvalues of the Laplacian of compact Riemann surfaces and Minimal submanifolds, Ann. Scuola Norm. Sup. Pisa, 7
(1980), 55-63.
Yano, K. [1]
The Theory of Lie Derivatives and its Applications, NorthHolland, 1957, Amsterdam.
[2]
Differential Geometry on Complex and Almost Complex Spaces, Pergamon, 1965, New York.
[3]
Integral Formulas in Riemannian Geometry, Dekker, 1970, New York.
[4]
Selected Papers, Amsterdam.
(Ed. by Obata), North-Holland, 1982,
Yano, K. and Bochner, S. [1]
Curvature and Betti Numbers, Ann. Math. Studies, 32 (1953), Princeton.
Yano, K. and Chen, B.-Y. [1]
Minimal submanifolds of a higher dimensional sphere, Tensor, 22 (1971), 369-373.
[2]
Pseudo-umbilical submanifolds in a Riemannian manifold of constant curvature, Differential Geometry, in honor of K. Yano, Kinokuniya, 1972, 61-71, Tokyo.
Yano, K. and Kon, M. [1]
Anti-invariant Submanifolds, Dekker, 1978, New York.
Yau, S.T. [1]
Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670.
AUTHOR INDEX
Abe, K. , 324
Adem, J., 195. 324 Asperti, A.C., 246, 324 Atiyah, M.F., 24B, 324
Banchoff, T.F., 324 Barros, M., 180, 181, 324 Bejancu, A., 172, 173. 176, 325 Berger, M., 25-. 100, 325 Besse, A.L., 325 Blair, D.E., 172, 173, 175. 325 Blaschke, W., 207, 22, 325 Bleeker, D., 296, 325 Bechner, S., 339 Borsuk, K., 158. 193, 235, 325
Calabi, E., 326
Cartan, E., 9 120, 127, 160. 161. 234, 326 Carter, S., 326
Cecil, T.E., 32E Chen, B.-Y., 115, 132, 141, 152, 171-176, 180, 183, 187, 193. 194, 196, 198, 201, 204, 206, 220-223, 226, 236, 239. 249. 255. 260. 269. 274. 276-279. 281, 283, 285. 289, 293, 296, 299, 300, 303-305.
307, 308- 313-316, 324-328, 339 Cheng, S.Y., 328 Chern, S.S., 2$. 84, 122, 157, 158. 162-165. 197. 200, 32.8
Dajczer, M., 246, 329. 330 do Carmo, M., 328
342
Author Index
Fells, J. Jr., 202, 328. 324 Ejiri, N., 207, 315. 316, 330 Erbacher, J., 330
Escobales, R .H . ,. 330 Fary, I., 158, 166, 330 Fenchel, W., 158, 186, 193, 235, 330 Ferns, D., 246, 324, 31Q
Gauduchon, P., 95, 100, 325 Gheysens, L., 269, 330 Goldberg, S.I., 84, 331 Gray, A., 316, 331 Guadalupe, IN., 247, 248, 331
Haantjes, J., 210, 3]. Heintze, E., 193, 331 Helgason, S.. 331 Hersch, J., 331 Hopf, H., 84 Houh, C.S., 220. 221, 222, 26,2, 321. 328. 331 Hsiung, C.C., 197. 293, 328, 331 Husemoller, D., 331
Jhaveri, C., 225. 3 8
Karcher, H., 193. 331
Klingenberg, W., 332 Kobayashi, S., 4, 30, 123, 328, 332, 332
Kon, M. , 33.9 Kuhnel, W., 332
Kuiper, N.H., 123. 16.5., 183, 234, 32Q. 332
Author Index Langevin, R., 166. 332 Lashof, R.K., 157, 158, 162, 163, 164. 165, 19L 200, 241- 328, 332 Lawson, 14-B- Jr., 169, 197, 198, 225, 248, 324, 333
Lemaire, L., 202, 328 Levi-Civita, T., 4.6
Li, P., 209, 235, 333 Lichnerowicz, A., 333 Little, J., 141, 154, 156, 241, 246, 333 Ludden, G.D., 328
Lue, Ham., 328 Maeda, M., 248, 333
Martinez, A., 316 Masol'cev, L.A., 295, 333
Mazet, E., 95, 1QQ, 322.5 McKean, H.P.
99 333
Meeks, W.H.. 166. 183. 332 Milnor, J.W., 22, 108, 158, 166. 333, 334 Minakshisundaram, S., 98 Minkowski, H., 223 Montiel, S., 328 Moore, J.D., 234, 334
Morse, M., 20, 21 22, 1fi4 Morvan, J.M., 334 Mostow, G.D., 334
Nagano, T., 115, 328. 334 Naitoh, H., 334
Nakagawa, H 334 Nash, J.F., 120. 1$Z, 334 Nomizu, K., 4 32. 12.3_. 332, 335
343
Author Index
344
Obata, M., 335 Ogiue, K., 132, 152, 328. 335 O'Neill, B., 73, 167, 3-15
Osserman, R., 335 Otsuki, T., 122, 236. 335
Palais, R.S., 76. $9 335 Patodi, V.K., 335
Perez, J.D., 3116 Pleijel, A., 98
Pohl, W.F., 333. Reckziegel, H., 336 Reeb, G., 22, 164 Reilly, R.C., 293. 295. 302, 3031 314, 336
Rodriguez, L., 246, 247, 248, 324, 331 Ros, A., 141, 180, 196, 267, 280, 281, 290, 311. 315.
316, 336 Rosenberg, H_, 332
Rouxel, B., 269. 3316 Ryan, P.J., 326
Sakai, T., 99, 3.316 Sakamoto, K., 141, 154. 156. 336
Sampson, H., 220 314
Santos, F.G., 316 Sard, A., 20
154
Shiohama, K., 184, 336 Simons, J., 33¢ Singer, I.M., 99, 324, 333
Smale, S . , 239. 241. 332, 331 Spivak, M., 337 Springer, T.A., 123
Author Index Sternberg, S., 24. 331 Sunday, D., 241. 3-32
Tai, S.S., 141. 145, 156, 33.7 Takahashi, T., 136, 138, 148, 307. 309. 312 Takagi, R., 184, 336 Takeuchi, M., 334. 337 Tanno, S., 332 Thomsen, G., 212, 221, 332
Urbano, F., 180, 181. 324
Vanhecke, L., 32B
Verheyen, P., 269. 328. 330 Verstraelen, L., 269, 334
Wallach, N.R., 138, 31 331 Weiner, J.L., 225. 296. 325 332 West, A., 32L White, J.H., 207, 212, 338 Willmore, T.J., 182, 113 184, 186, 225, 318 Wintgen, P., 240, 241, 242, 138 Witt, E., 1081 338 Wolf, J.A., 115. 334
Yamaguchi, S., 334 Yang, P.C., 315, 339 Yano, K., 132. 299, 328, 339
Yau, S.T., 209. 234, 235, 315, 33.31 335, 334
345
SUBJECT INDEX
Q - submanifold, 269 action, effective, 2.3
action, free, 23 adjoint, 79 affine connection, 46 allied mean curvature vector, 20 associated vector field, 56
associated 1-form, 56 asymptotic expansion, 4.8
betti number, 41 Bianchi identity, 55, 5B
CR-submanifold, 172 Cartan's lemma, 9
Cartan's structural equations, 5Q Casimir operator, 102 Cayley projective plane, 155 chain, 38
Christoffel symbols, 42 closed manifold, n codifferential operator, $Q cohomology group, 40 completely integrable distribution, 42 complex-space-form, 6$ conformal change of metric, 64 conformal Clifford torus, 344 conformal curvature tensor, 65 conformal square torus, 344 conformal Veronese surface, 344
conformally flat space, 66
348
Subject Index
connection, 46. 51 contraction, 6 convex hypersurface, 165
covariant differentiation, 46 critical point, 29 critical value, 29 cross-section, 24 cup product, 41
curvature tensor, 5Q curvature 2-form, 5Q cycle, 3.4
Dirac distribution, 46.
Einstein space, 54 ellipse of curvature, 245 elliptic operator, 8, a6 energy function, 2Q2 equation of Codazzi, 117 equation of Gauss, 117 equation of Ricci, 11S
equivariant immersion, 26 exact form, 4Q exotic sphere, 22 exponential map, 62 exterior algebra, @ exterior product, fl
exterior differentiation, 11 extrinsic scalar curvature, 295
fibre bundle, 23 finite type submanifold, 249
flat torus, 72
Subject Index
frame bundle, 22 Fredholm's operator, 84 Freudenthal's formula, 142 Fourier series expansion, 283 Forbenius' Theorem, 43 Fubini-Study metric, 24 fundamental 2-form, 677
Gauss-Bonnet-Chern's formula, 61 Gauss' formula, 1_Q9 Gaustein-Whitney's Theorem, 151 generic submanifold, 1.7.1
geodesic, 44
H-stationary submanifold, 214 H-variation, 214 harmonic form, 81 heat equation, 95 heat operator, 95 Hermitian manifold, 61 hessian, 100 Hodge-de Rham Theorem, 91, 9-2 Hodge-Laplace operator, 81 Hodge star isomorphism, 28 holomorphic distribution, 171 homogeneous space, 75
homology group, 221 3.9 Hopf fibration, 24 horizontal vector field, 73
index, 21
infinite type submanifold, 252 interior product, 10
349
350
Subject Index
Janet-Cartan's Theorem, 120
k-type submanifold, 252 Kaehlerian manifold, 61 Klein bottle, 73 knot group, 241 knot number, 241
Laplacian, $1 lattice, 22 Lie group, 23
Lie transformation group, 23 linear differential operator, 85 Lipschitz-Killing curvature, 152 locally finite covering, 2H locally symmetric space, 60
(M+,M-) -method, 115
mass symmetric submanifold, 274 mean curvature vector, 113. 114 minimal distribution, 174 minimal submanifold, 113 Morse's inequality, 22 Morse function, 21
Nash's Theorem, 121 non-degenerate function, 2.1
normal coordinates, 63
order of submanifold, 2 Otsuki frame, 236 Otsuki's lemma, 122
Subject Index
parallel translation, 49 partition of unity, 3Q Poincare duality Theorem, 93 projective space, 77
74,
25.
pseudo-Riemannian. manifold, 53
pseudo-umbilical submanifold, 132 purely real distribution, 121
quaternionic CR-submanifold, 180 quaterionic Kaehlerian manifold, 54 quaternion-space-form, 70
rank, 2fx Reeb Theorem, 22
regular point, 24 Ricci curvature, 54 Ricci tensor, 51 Riemannian connection, 55 Riemannian manifold, 53 Riemannian submersion, 73, 162 rotation index, 151
Sard Theorem, 2.0
scalar curvature, 51 Schur's Theorem, 51 second fundamental form, 111
self-intersection number, 233 simplex, 32 spectrum, 90 standard immersions, 138 Stokes' Theorem, 34 submanifold of finite type, 252
submanifold of infinite type, 252
351
352
Subject Index
submanifold of order [p,q], 2.52 submanifold of order p, 2552
submersion, 21 symbol of elliptic operator, 85, $fi
symmetric space, 15
tension field, 202 tensor, 1 tensor product, 3
tight immersion, 154 torsion tensor, 50 total differential, 1.61
total mean curvature, 1fl1
total tension, 202
totally geodesic submanifold, 101 totally real distribution, 172 totally umbilical submanifold, 113 2-type submanifold, 260
About the Author
Dr Bang-yen Chen is Professor of Mathematics at Michigan State University. He has held visiting appointments at many universities,
including the Catholic University of Louvain. National Tsinghua University of Taiwan, Science University of Tokyo, University of Notre Dame, and University of Granada. Dr Chen's research interests focus on differential geometry, global analysis and complex manifolds. He is the author of numerous articles and two books Dr Chen received his B.S. degree in 1965 from Tamkang University. his M.S. degree in
1967 from Tsinghua University and his Ph.D. in 1970 from the University of Notre Dame. He is a member of the American Mathematical Society.
9971-966-03-4 pbk
