G-total curvature of immersed manifolds

Full text

J. DIFFERENTIAL GEOMETRY
7 (1972)
371-391
G-TOTAL
CURVATURE
OF IMMERSED MANIFOLDS
BANG-YEN
CHEN
Given an immersion x: M
—>
Em of a bounded manifold M of dimension n
in
a euclidean space Em of dimension m, we define what we call the G-total
curvature with respect to a
given
vector-valued function g on the normal
boundle Bv as the integral over Bv of g times a power of a general mean
curvature, i.e., \
giK^dV
Λ dσ. We also define the G-total absolute curva-
tures in a similar way. The main purpose of this paper is to
give
the relations
between different G-total curvatures or G-total absolute curvatures depending
on
g, i and m,
first
for a
fixed
immersion and later for different immersions.
In
particular, our results generalize many well-known results in differential
geometry such as Gauss-Bonnet's formula, Chern-Lashof's theorems,
Minkowski-Hsiung's formulas, etc.
1.
Definitions
Throughout this paper, a bounded manifold means a compact manifold with
or
without smooth boundary. A closed manifold is a (compact) bounded mani-
fold without boundary. Let M be a bounded manifold of dimension n, and
x:
M
—>
Em an immersion of M into a euclidean space Em of dimension m.
Suppose that Em is oriented. By a frame P,e19
-—9em
in the space Em we
mean
a point P € Em and an
ordered
set of
mutually perpendicular unit
vectors
*i> * #
5
£m
with
an
orientation
coherent
with that
of the
space
Em. Let F(Em)
be the set of all
frames
in the
space
Em, and F(M) be the set of all (ortho-
normal)
frames
in M
with respect
to the
induced metric
on M.
To
avoid
confusion,
we
shall
use the
following
ranges
of
indices
throughout
this
paper unless
otherwise
stated:
1 <
/,/,Λ,
••• < n; n + 1 < r9s, t, < m; 1 < A, B, C, < m .
In
F(Em) we introduce the
1-forms
ΘA, ΘAB by
Received June
26, 1970, and, in
revised form, December
27, 1971.
This paper
is a
condensation
of the
author's
Ph. D.
dissertation submitted
to the
Graduate School
of
the
University
of
Notre Dame, June
1970. The
author
wishes
to
express
his
hearty thanks
to
his
thesis advisor Professor
T.
Nagano
for his
constant encouragement
and
valuable
suggestions.
He is
also
very
thankful
to
Professors
Y.
Matsushima
and T.
Otsuki.

372
BANG-YEN
CHEN
(1.1) dx=Σ0ΛeA9
deA=Σ0ABeB,
OAB +
ΘBA
= O.
Since
(1.2) d(dx)
= O,
d(deA)
= O,
from (1.1)
we
have that
(1.3)
dθA = Σ
ΘB
Λ ΘBA ,
dfl^
=
ΣΘAC
A ΘCB ,
where
Λ
denotes the exterior product.
Let BΌ denote the bundle
of
unit normal vectors
of
x(M)
so
that
a
point
of
Bv
is a
pair (P, e) where
e is a
unit normal vector
at
x(P). Then
Bυ is a
bundle
ot
(m — n —
l)-dimensional spheres over
M
and
is a
(smooth) manifold
of
dimension
m — 1.
Let
5 be
the
set of
elements
b =
(P, e^
,
em) such that
(P,
ex,
,
en)
6
F(M)
,
(jt(P),
ex,
,
e
J e
F(E-)
,
where
we
have identified e* with dx(ei). Then
B
—>
M
may
be
regarded
as a
principal bundle with fibre 0(ή)
x
5O(m
— ή),
and jc:
£ -»
F(£m)
is
naturally
defined
by
x(b)
=
(x(P),
e19 ,
eTO). Let ω^, ω^β
be
the induced
1-forms
from
ΘA, ΘAB
by
the mapping
x.
Then
we
have
ωr = 0,
and
ω1? ,
ωTO
are
linearly
independent.
Hence
the
first
equation
of
(1.3)
gives
Σωι A ωίr = 0. By
Cartan's
lemma
we
may write
(1.4) ωίr=
ΣΆjωj
. AlJ =
Ar
jί
.
The
eigenvalues kx(P,
er), ,
kn(P,
er) of
the symmetric matrix
(A^)
(which
is called
the
second fundamental form
at
(P, er))
are
called
the
principal
curvatures
of M at
(P,
er).
The
ί-ίΛ
mean
curvature
Ki{P,er)
at
(P,
er) are
defined
by
the elementary symmetric functions
as
follows:
(1.5)
(") W,er) = Σ UP,er) -UP,er) , i - 1, ,n ,
where
ί^j =
n\/[i\
(n
-
/)!].
In
the following,
let
dF
=
ωx
Λ Λ ωn
and
rfσ =
ωm>n+1
Λ Λ
ωTOfm_i.
Then
dF
is
the volume element
of
M, and
dσ is a
differential
(m — n — 1)-
form on
BΌ
such that
its
restriction
to a
fibre 5^"n"1
of Bυ
over
P € M is the
volume
element
of
S^'71'1.
Furthermore,
dσ Λ dV can be
regarded
as the
volume element
of Bv
(for the detail,
see
[10]).
Let
V be a
finite dimensional vector space over R, and
let
(1.6)
g:Bv->V
be
a
F-valued continuous function on the normal bundle
Bv.
The integral

G-TOTAL
CURVATURE 373
(1.7)
GfaP^m)
= J
is called the i-th
G-total
curvature
of rank m at P
with
respect
to g, and the
integral
(1.8)
Γ,(jt,g,m)
-
JGi(x,P,g,
M
m)dV
is called the i-th
G-total
curvature
of rank m
with
respect
to g if the right hand
side of (1.8) exists. The integral
(1.9)
K*(x,P,g,m)
=
sm-n-i
is called the i-th
G-total
absolute
curvature
of rank m
with
respect
to g at P,
and
the integral
(1.10)
TAfa&m) = Jκ*(x,P,g,m)dV
M
is called the i-th
G-total
absolute
curvature
of rank m
with
respect
to g if the
right hand side of (1.10) exists.
In
this paper, let X denote the Em-valued function on Bυ which maps (P, e)
€ Bv onto x(P), and e the
£m-valued
function on Bυ which maps (P, e) e Bυ
onto
e. We also denote by X the position vector field on M in Em.
The
G-total absolute curvatures have been studied previously by Chen [4],
[6] and Santalό [17] for arbitrary i-th G-total absolute curvatures, and by
Chern-Lashof
[10], Chen [2] and many others for the last G-total absolute
curvature.
For the relations between i-th G-total absolute curvatures and
integral geometry, see Chern [8], Santalό [17].
2.
Elementary
formulas
Through
a
point
in Em, let
i?ls
,
ι?m_15
i bem
vectors
in Em, and let
Vι
X X
Um-i denote
the
vector product
of the m — 1
vectors
vλ, ,
tfm_i
Then
(2.1)
V iVί X ••• X Vm-d =
(-l)m~ΊlM>i> •• ,I?m-il
>
where
\v, ι?15 ,
ϋm_i| denotes
the
determinant
of v,
Όι,
,
ι?m_i From
(2.1)
we have
(2.2)
e, X - X eA X X em - (-l)m+ΛeA ,

374
BANG-YEN
CHEN
where the roof means the omitted term. In the following, let < , ) denote the
scalar product in Em, and x the combined operation of the vector product
and
the exterior product. We
list
a few formulas for later use:
(2.3) d*x = d2eA = 0 ,
(2.4) dx X >< dx x en+1 X X er X X em = n\ (-1)-+'erdV ,
n
\der,
, der, dx, , dx, en+1, , έ?w|
(2.5) "- I * ^—'
=
(- l)*π! Kt(P, er)dV (i not summed) ,
(2.6) p(P, e) = x(P) e , K0(P, e) = 1 .
In
the following, if there is no danger of confusion, we shall simply denote
Ki(P, e) by Ki9 em by e and Af- by Ai5.
3.
Mean
curvature
form
Let
(3.1)
θ = Σ
(-1)*~X
Λ
ΛώiΛ
Λ
wnet
.
Then
θ is a well-defined vector-valued (n — l)-form on M, and is called the
mean
curvature
form
of the immersion JC: M
—>
Em. Since we have, from a
direct computation, that
(
1Λm-l
(3'2)
θ =
(n-l)\
d-X * "t-*dx * en+ι * ' ' * * e" '
by taking exterior derivative of (3.2) we obtain
(3.3) dθ = nHdV ,
where H = (l/ri) Σ Ar
uer is called the mean curvature vector. If the mean
curvature vector H = 0 identically on M, then M is called a minimal sub-
manifold of Em. From (3.3) we see that
Observation.
M is a minimal
submanifold
of Em
when
and
only
when
the
mean
curvature
form
θ is
closed.
Moreover, by (3.3) and Stokes' theorem, we have
Proposition
3.1. Let x: M
—>
Em be an
immersion
of an n-dimensional
bounded
manifold
M in Em. Then
(3.4)
n JHdV = jθ ,
where
dM
denotes
the boundary of M.

G-TOTAL
CURVATURE 375
Proposition 3.2. Under the hypothesis of Proposition 3.1, we have
(3.5)
nv(M) + n J<X, H}dV =
where v(M) and X denote the volume and position vector field of M,
respectively.
Proof.
By taking exterior derivative of ζX, ©> and applying (3.3) we
obtain
(3.6)
d(X9 ©> = ndV + n(X, H}dV .
Integrating
both
sides of (3.6) over M and applying Stokes'
theorem,
we
obtain
(3.5).
Remark
3.1.
Proposition
3.2 was
obtained
by Hsiung [13] for n = 2 and
by
Chern-Hsiung
[9] for closed M.
Corollary 3.3 (Chern-Hsiung [9]). There exist no closed minimal sub-
manifolds
in a euclidean space.
Corollary 3.4. // M is a minimal submanifold of Em, then
(3.7)
nv(M)
= J<X,Θ>.
dM
These
two corollaries follow immediately
from
(3.5).
By Corollary 3.4 we
have
Corollary 3.5. Let M and M' be two bounded minimal submanίfolds of
Em
such that (a) there exist two neighborhoods U and JJf of dM and dMf
respectively such that U = U' and (b) dim U = dim V = dim M = dim Mf.
Then
v(M) =
v(Mf).
4.
Differential
formulas
Let
α = Σ aιei be a
smooth
vector field on M.
Then
(4.1)
da= Σ (daj + Σ
aj^ij)^j
+ Σ
<^rer
,
and
therefore
d<a, θ>= Σ
(-Όj-\daj
+ Σ Wij) Λ ωγ A Λ ώs Λ - Λ ωn .
Thus,
if we put da5 = Σ
(aj)k^k
and ωtj — Σ Γj
ikωk,
then
(4.2)
d(a,θ}
= (div a)dV ,
where
diva = Σ (aj)j + Σ
aiΓίj
^om (4.2) follows immediately
Proposition 4.1. Under the hypothesis of Proposition 3.1, we have

376
BANG-YEN
CHEN
(4.3)
for any
tangent
vector
field
a on M.
Let / be a smooth function on M. By
gradf,
we mean
gradf
= Σ /^,
where df = Σ
/«<*>*•
Since
(4.4) d(fθ) =
(gradf)dV
+
nfHdV
,
from Stokes' theorem .we obtain
Proposition 4.2.
Under
the
hypothesis
of
Proposition
3.1, we
have
(4.5) /i J/JWK + J
(g/vϊtf
βdF = jfθ .
Let g be a smooth function on the normal bundle Bv. Put dg = J]
Σ
Srθ)mr
and Vg = 2 g^. Then we have
Lemma 4.3.
Under
the
hypothesis
of
Proposition
3.1, we
(4.6) d(gβ Λ dσ) =
(Γg)dV
A dσ + ngHdK Λ rfσ .
Proof.
By taking exterior
derivative
of gθ Λ dtf and applying (3.3), we
obtain (4.6) immediately.
There
exists
a self-adjoint linear transformation A of the tangent space
TpiM) of M at P into
itself
defined by
(4.7) Aet= -
where (Aί3) denotes the second fundamental form at (F, e). It
follows
that
(4.8) A(dx) =
where (J^)ί is the tangential component of de. Let A(j)(dx) denote the tangent
vector obtained from dx by applying A repeatedly / times, and * the Hodge
star operator defined by
(4.9) *(Σ
fiCOiβi)
= Σ (-l)*"1/^! Λ
ΛώtΛ
Λ
ωnet
.
For
convenience we put UQ = dx and Uj = A(j)(dx), j = 1, 2, .
Lemma 4.4. Le* e = em. Then
': en+1 X X en
(4.10) ^"^ i
t n
δ \i -
I.

G-TOTAL
CURVATURE
377
This lemma
can be
proved
in the
same
way as
Lemma
2.1 was
proved
in
[1],
so we
omit
the
proof here.
Lemma
4.5. Let
Ji
= ex de
>< StdeXdxZ
- - * dx ,
(4.11)
m-n
+
i-l n-i-1
/
=
0,1, ,/i
— 1 .
Then
Δi
= _(m _ n + i- 1)! (n - i - 1)! (-1)' Σ L " h)Ki-**Uh A da
(412)
+ nKm-n + f-DI ^
(_1)i+s+^+idF
Λ ^
(l + 1)! s
=
n
+
l
Λ
Λ
ώTOtί
Λ Λ
ωTO,m_Λ
.
This lemma
can be
proved
by a
direct computation
of the
left
hand side
of
(4.12)
we
omit
the
proof.
Lemma
4.6. Let
(4.13)
^ =
φΛ<I,^>, XA
=
<X,eA>,
i = 0, 1, ,n - 1 .
=
(/w-n + ί-
l)!(/i-i-
1)! Σ
1.14)
,. n ΛKt_h Σ
xμjAiΔ
Π ^h
l)t)dV
Λ dσ
+
(iy: ;
0
+ 1) !
Proo/.
By
(4.8)
and
(4.9),
we
have
(4.15)
Ut= Σ (-
and therefore
(4.16)
*Ui = Σ
(-lV0+ί+1(Π
^^-J^iΛ
Λώi0Λ
ΆωneJt
By Lemma
4.5, we
obtain

378
BANG-YEN CHEN
i
^
V \i —
^A(
Π
A
=
l \k
=
k
l }
-n-H-1)!
/
£ W Λ
+
(!)^
0+1)!
From
this
we can
easily derive (4.14).
Lemma
4.7.
dp
A Δi - (m - π + i - 1)! (n - i - 1)! £ Σ (-
Λ
=
0
jo," Jh
=
l
*-» Σ
*Λ^«.(Π/Λ-IΛ)^
Λ da
0 + 1)!
This
lemma
can be
proved
in the
same
way as we
prove Lemma
4.6.
Lemma
4.8.
d(pι-\X,
Jt» =
(I
-
ί)pι-%
(4
18) - (- lyK 'Cm-n + i-ΌV-^ Λ dσ
Proo/.
Since
</«
-k
• •
X de % dx -k
• •
:><
dx
(4
19) m-n
+
i-l »-i
'(+
i
(
z!
(4.20)
nl(m
by using (4.11), (4.13), (4.19)
and
(4.20)
we can
prove (4.18) without
difficulty.
Lemma
4.9.
n\
ί (n — ί)(m — n + ΐ) tJ, , ni-i^
-
(I - Dp'-'K, Σ
xμkAjk}dV
A dσ

G-TOTAL
CURVATURE
379
n-l
(4.21)
+ " (/ -
IV-2
Σ
(*s)2Ki+1dV
A
dσ
V
~Γ Ll s
=
n
+
l
(m
_ n + ί _ i) f (n _ / _ i)!
h*Uh)>}
A dσ .
Proof.
By
Lemma
4.4,
(2.5)
and
(3.1),
we
have
h=i
V h) % \ι I h=Q \ι nj
X
d ><
SI "V
\/ m \f /7f* \f
Si
0 Sf . \S
SI
0 \f 0 \/ ... Sc 0
n-i-1 ί
and
therefore
Σ
,
i!
(n - i - 1)!
( X) jΛ AJAJA AjA A A
+
dx
X
XdxXdeX ••• XdeXen+1X •••
Xe,
iΠi1)1
n-i
=
-n
ifjKidV
+ h)(n
-
i)K4V
=
-iί"
On
the
other hand, from (4.12), (4.13)
it
follows immediately
(m
—
n + ϊ
— 1)!
(n
—
i
— 1)!
u +
1/
s=n+i
s ι+ι
Substituting
the
right side
of
the
above
two
equations
in
the
following equation
and
simplifying
the
resulting equation
by
using (4.18)
we
can
easily reach
(4.21):
ΛΣ
(/ 1
^{dip^K^ζX, *Un))
-
pι-\X,
d(Kt_h*Uh)y}
A
dσ
i
/ n \
—
V \U1
λλnι~2K
ήn A /Y
*TT
\
7ι =
l\ /
+
pt-'K^dX,
*Uhy)
A dσ .
Lemma
4.10.
Let

380
BANG-YEN
CHEN
(4.22)
ψ =
Σ1 (-DW
Λ
ωm,n+1
Λ Λ
ώm>s
Λ Λ
ωm,m_A
s =
n
+
l
Then
(4.23)
-
(/
-
ί)p1-2
S Σ
(xs)2Kί+1dV
A
dσ
,
ί
=
0,1,-•.-,/!-
1 .
Proof.
By using (dx, Ki+1ψy
=
0, we have
d<pι-ιKuιX,T>
-
Pι-\X,d(Kί+1Ψ)>
= (/
-
i)pι~2κuidp
A
(x,wy +
pι-ικί+ι<dx,ψy
Lemma
4.11.
n-l
n / n \ n
Σ
i
V (
1
\i
—1 /
n \V V1
i-U
f-i n - i - 1
"-*-1
A
(4.24)
ι-°JO'-'^-1
Proof.
For simplicity, we choose the principal frame with respect
to
e = em, so that
ωmί
=
—kiίύi (i not summed)
.
By
a
direct calculation we can
easily
obtain (4.24).
Similarly we can prove
Lemma
4.12.
n-l
I yj \
t_Λ~U \n-i-ί)K"-^^
(4.25)
5.
Integral
formulas
and
their
applications
Theorem
5.1. Let
x:
M
—>
Em
be
an
immersion
of an
n-dimensίonal
bounded
manifold
M in Em. Then we have
(5 D
_
(/
_
i)rl(jc>
Σ
xj^^p'-2,1)
=
I
p'-'^ΛΓ,
θy^dσ,
dBΌ

G-TOTAL CURVATURE
381
for
all ί = 0, 1, , n — 1 and an
integer
I.
Proof. By
Lemma
4.5 and
(4.22),
we
have
(«-«
+ »-
1) !(n-i-
1)! VI l' +
By
first
taking exterior derivative
of
(5.2), using (4.19)
and
applying Lemma
4.3,
and
then taking scalar product
of X
with both
sides
of
the resulting equa-
tion
and
multiplying
by pι~\ we
obtain
(m
- n + i)(.
?\)pιKi+1dV
A dσ =
[n\pι-ι<X,VK>>dV
A dσ
+
Σ (i 1
^Pι-\x,d(κi_h*uh
A
dσ)}
+ (.
Substituting (4.21), (4.23)
in the
above equation
for the
last
two
terms
and
simplifying
the
resulting equation
by
using (4.12), (4.13), (4.22)
we can
easily
obtain
ip^Kt-d
- Dp1"'*, Σ
XjX*Ai*W
Λ da
(rn
- n + / - 1)! {n - i - 1)!
h
χ .. 1
J^-^.^AΓ,
*Uh
A
dσ))
Integration
of
both
sides
of the
above equation
and
application
of
Stokes'
theorem
give
immediately (5.1).
Theorem
5.2.
Under
the
hypothesis
of
Theorem
5.1, we
have
-
1) Σ Σ
(-I)*"1 L
"
.7=1
(m - n +
/)/.
£
λτuι(x,pι,
1) + (n - ίjh

382
BANG-YEN
CHEN
(m
— » + i — 1)! (« — i — 1)! J
i
=
0,1, •••,/!-
1.
This theorem
follows
from Lemmas
4.6 and 4.8.
Theorem
5.3.
Under
the
hypothesis
of
Theorem 5.1,
we
have
(m-n +
(5.4)
+
(m
— n + i — 1)! (n — i — 1)!
Proof.
Substituting (4.17), (4.19)
in
d(pιJt)
=
pldAt
+
lpι~ιdp
A Δ, ,
we
can
easily obtain
'-1
Σ
m-1
Adσ-lp1-^,
MKJ Σ
x*.)dVΛ
-=
(-l)i+ίd(pιJJ/[(m - ft + i - 1)! (/ι - i - 1)!] .
Integrating both sides
of the
above equation
and
applying Stokes' theorem,
we hence have (5.4).
Theorem
5.4.
Under
the
hypothesis
of
Theorem 5.1,
we
have
(I
-
l)Tn(x,pι-\X,X>,
1) +
nTn_Ύ{x>Pι-ι>
D
(5*5)
= (m + I -
2)Tn(x,
p\ 1) + ("J0^*
Proof.
This theorem
follows
from Lemma 4.11, Theorem
5.2 for / = π
—
1
and
the following identity ΣA=1
*A*A
= XX - P2
For
/ = 1,
Theorem
5.2
reduces
to
Corollary
5.5. If M is
closed,
then
we
have

G-TOTAL CURVATURE 383
(i + Wax, 1,1) + (m - n +
i)Ti+ι(x,
p, 1) = 0 ,
i
= 0,1, ..-,/! — 1 .
Remark
5 1. If the codimension m — n — 1, then (5.6) are Minkowski-
Hsiung's formulas [12].
If / is odd, then Gι(x,P, 1,1) = 0; if / is even, then Gt{x, P, 1,1) depends
only on the Riemannian structure of M with the induced metric (see Remark
8.2). Hence from Corollary 5.5 we obtain
Corollary
5.6. // M is
closed,
then
the ί-th
G-total
curvatures
Tt{x, p, 1)
for all ί = 1, -,n
depend
only
on the Riemannian
structure
of M
with
respect
to the
induced
metric.
In
other
word,
Tt(x, p, 1) is an
isometric
invariant
for all i = 1, , n.
From
Corollary 5.5
follows
Corollary
5.7. // M is a
complete
submanifold
of Em
with
Gλ(x, P, p, 1)
=
0
everywhere
on M,
then
M is not
compact.
Putting / = 0 in Theorem 5.3 we obtain
Corollary
5.8. // M is
closed,
then
Tt{x, e,l) = 0 for all i = 1, , n.
Corollary
5.9. // M is
closed,
then
(5.7) (m + / - l)Tn(x, pιe, 1) = Tn{x,
pι~'X,
1) .
Corollary 5.9
follows
from Lemma 4.12 and the identity X — pe —
2-χA=l ΛA^A'
Corollary
5.10. // M is
closed,
then
(5.8) nUx, H, 1) + T0(x, VKt, 1) = 0 , i = 0,1, . ., n - 1 .
In
particular,
if Gt(x, P, 1,1) is a
constant,
then
(5.9)
Ux,rκi91)
= 0 , i =
0,1,
..,n- 1 .
The
first
part of this corollary can be obtained by applying to (5.1) for
/ = 1 a translation x
—>
x + c where c is any constant vector in Em, and the
second part
follows
from Proposition 3.1 and (5.8).
Corollary
5.11. Let M be an
n-dimensional
oriented
closed
submanifold
in Em
such
that M
does
not
contain
the
origin
and the n-th mean
curvature
Kn(P, e) is
nonnegative
everywhere
on Bυ. Then
(5.10) nTn_x{x, p~\ 1) > (m - 2)Tn(jc, 1,1) .
Proof.
This corollary
follows
from Theorem 5.4 for
Z
= 0 and the assump-
tion
Kn(P, e) > 0.

384
BANG-YEN
CHEN
6.
Gauss-Bonnet's
formula
In
this section, we shall assume that M is an n-dimensional oriented closed
manifold imbedded in Em.
Proposition
6.1. Let a19 -,ah be h nonnegative integers, and αlv , ah
be
h fixed vectors in Em.
Then
Π
3=1
(6
(6
.1)
Proof.
.2)
=
Put
β =
+
<xhTnL
(m
+ aλ -
m-l
Σ(-DJ
/ι-l
^,
Π
+
i Λ
αft
- l)Γn(x,
•
Λ ώmιA I
Q Σ ,i m,A Λ
-4
=
1
Then
dex -
XdeXe
=
(m-2)\
(-l^-
m-2
On
the other hand, from (4.19) we have
(6.3) de X X de = (-l)w(m - 1)! KnedV A da .
Therefore
(6.4)
(-ΐ)ndQ
= -(m -
l)KnedV
A da .
Moreover, we can prove that
(6.5) (-
\)n<X,
de) A Q = (X - <X, e}e)KndV A da ,
(6.6) (- l)»<α, de) A Q = (a - <α, ^>? ri^dK Λ da ,
where α is a
fixed
vector in Em. Hence by taking exterior derivative of
<α1,e>βl
<αΛ_1,e>βfc-ι<^«>βfcβ and applying (6.3), --.,(6.6) and Stokes'
theorem,
we can obtain (6.1).
Proposition
6.2. Let a be a
fixed
vector
in Em
perpendicular
to
a2> ''
>«Λ-I
Then
=
(m + «! + + ah — l)Tn(x, <«!,

G-TOTAL
CURVATURE 385
By taking scalar product of (6.1) with a, we obtain (6.7).
Proposition
6.3. // aλ is a
fixed
unit
vector
in Em
perpendicular
to
a2, - - , ah, and aλ is a
positive
even
integer,
then
(6.8) Tn(x, <βl, ey> <ah, eY\ 1) = γTn(x, <α2, «>'• <*„ e>a\ 1) ,
where
(6.9) f — 2,cm+ai+...+ah_1/i<caicm+a2+...+ah_1) ,
and
ck = 2πia+1)
/Γ(^(k
+ 1)) is the area of the
unit
k-sphere.
Proof.
Setting ah = 0 and a = ax in (6.7) we readily obtain
(6.1(J)
=
(m + ttl + + ah -
2)Γn(x,
<α1?
e>αj
••-
<αΛ,
e>α%
1) .
Repeating
(6.10)
for \ax — 1
times
thus
gives
(6.8).
Proposition
6.4. Let χ(M)
denote
the Euler
characteristic
of M. Then
(6.11) ΓΛ(JC,l,l) = cTO.1χ(Af).
Proo/.
If dimM = π is odd, then we have
Gn(x,P,
1,1) = 0, so that
Tn(x, 1, 1) = 0. On the other hand, by the Poincare duality, we have χ(M)
=
0. Thus we obtain (6.11). Now assume n to be even. If the codimension
m — n is odd, then the normal bundle Bυ has dimension m — 1. Since Bv is
closed and oriented, from Gauss-Bonnet's formula we have
(6.12)
Tn(x,l,l)
=
icm_lX(Bv).
Since Bv is a bundle space of (m — n — l)-dimensional sphere over M, we
have χ(M) = χ(Sm-n"1)χ(M) = 2χ(M). Hence (6.12) reduces to (6.11). If the
codimension m — n is even, then we define an immersion x: M -+ Em+1 by
JC(P)
= X(P) for all P in M. By a direct computation, we obtain cmTn(x, 1,1)
=
cm-iTn(x, 1, 1) = Cm-iCm%(^)> which implies (6.11).
The main purpose of this section is to prove the
following
generalization of
Gauss-Bonnet's formula.
Theorem
6.5. Let a19 , ah be h
nonnegative
integers,
and a19 , αΛ be
h
orthonormal
vectors
in Em. Then
(6.13)
Γ»fc(βl,ί)
....<Si,{) M)= MM) if ai,.. , a, are even,
10,
otherwise,
where
(6.14) ί =
2hcm+ai+..,+ah_1/(cai
.. cj .

386
BANG-YEN
CHEN
Proof.
If a19 , ah are all
even, then
by
applying Proposition
6.3 for h
times and using (6.11)
we
obtain (6.13).
If
at
least one
of a19 , ah is
odd, then without
loss
of
generality
we can
assume
aλ to be
odd. Application
of
(6.10)
for \{aλ — 1)
times thus
gives
,,
1C,
Tn(x,<fi19ey*-
<ah9ey*,l)
(ΌΛJ)
=
cTn(x,
<βl,
e><«2,
e>"°
<αft,
*>"»,
1) ,
where
c is a
constant. On the other hand,
by
Proposition
6.1 we
have
(6.16)
Σ
«Λ(«, <*,«>•*•••<«*,
>- i,l)
=
(m + a2 + + ah —
1)TW(JC,
<α2, e>"2 <αΛ,
e>"Λe,
1) .
Thus
by
taking scalar product
of
(6.16) with
ax we
obtain
(6.17) Tn(x, <Λl,
*><α2,
e}"* <αΛ,
ey\ 1) = 0 .
Combination
of
(6.15) and (6.17) hence
gives
(6.13).
Remark
6.1. Theorem
6.5 is
the well-known Gauss-Bonnet formula when
ax
= ah = 0,
and
was
proved
in [3]
when
h = 1
and
m — n = 1.
7.
Immersions
with
Lipschitz-Killing
curvature
> 0
For
an
immersion
of an
π-dimensional manifold
M in Em, the
n-th mean
curvature KW(P, e)
is
also called the
Lipschitz-Killing
curvature.
In
[10],
S. S.
Chern
and
R. K.
Lashof studied
the n-th
total absolute curvature
of
rank
1
with respect
to 1, i.e.,
TAn(x, 1,1),
and
proved
the
following
interesting
inequality
for
closed
M:
(7.1) TAn(x,
1,1) >
β{M)cm_x
,
where β(M)
=
max
{Σΐ=o
dim Ht(M;
F): F
fields},
and
H^M.F) denotes
the
i-th homology group
of M
over
F. If we
denote
the i-th
betti number
of
M
by
bi(M), then
it is
obvious that β(M)
>
Σί=obi(M).
In
this paper,
an
immersion
of an
n-dimensional closed manifold
M in Em is
called
a
minimal
imbedding
if
T^O*:,
1,1) =
β(M)cm^.
In
the following,
let
(7.2) J(P)
=
max{Kn(P,
e);ee
Srn~1}
,
(7.3) μ(P)
=
min
{£W(P,
e):ee
ST71'1}
,
^+
=
{(P,
e)
€
5,: Kn(P,
e)>0},
A_
= {(P,e)eBυ:Kn(P,e)<0}9
(7.5) Jl+(P)
=
max {J(P),
0} ,
/.-(P)
=
min
W«, 0} ,

G-TOTAL CURVATURE 387
(7.6)
KM) = i\β(M) -
\
where
S™~n~ι
denotes
the
unit
(m — n — l)-sphere of
unit
normal
vectors to
x(M)
at x(P) in Em. We call λ and μ the principal curvature and secondary
curvature of M in Em. If is clear
that
M has no
torsion
when and only when
t(M)
= 0.
Proposition 7.1. Lei M be an n-dίmensional oriented
closed
manifold
imbedded in Em. Then
Σ
(7.7)
(λ+dV
> (t(M) + Σ b2ί
(7.8)
(μ-dV < - [t{M) + Σ
Equality
sign
of (7.7) /zoWs
W/Z^AZ
tffld c?«/j >v/zew /Λe codimension m — n = 1
and
x:
M—>
Em is a minimal imbedding. Moreover, equality
sign
of (7.8)
holds when and only when either (a) dim M = n is even and the Lipschitz-
Killing curvature Kn(P, e) > 0 everywhere, or (b) the
codimension
m — n = 1
fl/zd
x:
M
-+
Em is a minimal imbedding.
Proof.
From
Theorem
6.5 it follows
that
(7.9)
(κn(P, e)dV
Λdσ+
fκn(P,
e)dV A da = Σ
{-l)%{M)cm_x
.
J
J i
=
0
A+
A-
On
the
other
hand,
by (7.1) and (7.6) we have
JKn(P,
e)dV
Ada-
Jκn(P,
e)dV
A da
(7.10)
A+ A-
It
Combination
of (7.9) and (7.10) yields
(7.11)
(Kn(P,
e)dV
Λdσ>
Uht) + Σ I
J \ ί
=
0
A
+
(7.12)
JKn(P,
e)dV
A da < -
(t(M)
+ |
A-
Therefore
by
(7.2), (7.3),
(7.4) and (7.5) we
obtain
(7.7) and
(7.8).
Now
suppose
that
equality sign of (7.7)
holds.
Then
the inequalities
(7.10),
(7.11)
and
(7.12) are actually equalities, so
that
x: M -> Em is a
minimal
imbedding.

388
BANG-YEN
CHEN
Next
suppose that the codimension m — n > 1. It is easy to see that if λ(P) > 0
at
P
<ε
M, then Kn(P, e) = λ(P) for all (P, e) e Syn~ι. In particular, this implies
that
dimM = n is even. Since the set {(P,e) εBυ: the second fundamental
form at (P, e) is positive definite} is of positive measure, by choosing a point
(P,
e) in this set we have λ(P) > 0. Thus we obtain Kn(P, e) = λ(P) for all
e €
S^~n~ι.
On the other hand, by definition we see that the second fundamental
form at (P, —e) is negative definite, and the continuity of the second funda-
mental
form implies that the Lipschitz-Killing curvature Kn(P, e) — 0 for some
points
in
S^'71'1.
Since this is a contradiction, we get m — n = 1. Conversely,
if m — n — 1 and x: M
—»
Em is a minimal imbedding, then the equality sign
holds in (7.11) and (7.12). On the other hand, Kn(P, e) = λ+(P) on A+ and
Kn(P,e) = //-(P) on >ί_: Moreover, A+ = {PzM: λ+(P) Φ 0} and A_ =
{P e M: /r(P) ^ 0}. Consequently, the equality sign holds in (7.7) and (7.8).
Now
suppose that the equality sign of (7.8) holds, and the Lipschitz-Killing
curvature Kn(P, e) < 0 for some points (P, e) in Bv. Then μ~(P) < 0 for some
P
in M, and Kn(P, e) = /i~(P) for all (P, *) 6 Syn~l whenever μ~(P) < 0. This
is impossible by the continuity of the second fundamental form on the fibre
Sp'71'1
if the codimension m — n > 1. Thus we get m = n + 1. On the
other
hand, from the equality of (7.8) and the inequality of (7.10) it
follows
that
the equality sign holds in (7.11) and (7.12). This implies that the immer-
sion of M in Em is a minimal imbedding. Consequently, either the Lipschitz-
Killing curvature is nowhere negative, or m = n + 1 and
JC
: M
—>
Em is a
minimal
imbedding. In the
first
case, we have t(M) = 0 and b^M) — 0 for all
odd
/. Thus (a) if Kn(P,e) is nowhere negative, then by the inequality (7.8)
we have t(M) = 0 and b^M) — 0 for all odd /, and therefore by (7.3) and
(7.5) we get the equality sign of (7.8) and (b) if m = n + 1 and
JC:
M —» Em
is a minimal imbedding, then the equality sign of (7.8)
follows
immediately
from the equality sign of (7.10) and the definition of μ. This completes the
proof of the proposition.
Theorem
7.2. Let x: M
—>
Em be an
imbedding
of an
n-dimensίonal
oriented
closed
manifold
M in Em. (a) The
Lipschitz-Killing
curvature
Kn(P, e)
> 0
everywhere
if and
only
if (i) M has no
torsion,
(ii) all
odd-dimensional
bettί
numbers
of M vanish, and (iii) the
imbedding
x\ M
—>
Em is minimal.
(b) // the
Lipschitz-Killing
curvature
Kn(P, e) > 0
everywhere,
then dim M is
even, and
either
dimM = 0 or the
codimension
m — n = 1, M has no
torsion,
and x(M) is a convex
hypersurface
in En+1.
Proof,
(a) If the Lipschitz-Killing curvature Kn(P, e) > 0 everywhere, then
μ-(P)
= 0. Thus by Proposition 7.1, we obtain t{M) = 0 and b^M) = 0 for
all odd /. Moreover, A_ = 0. These imply that
TAn(x,
1,1) = Tn(x, 1,1) = χ(M)cm^ = i8(A0cm_1 ,
i.e., the imbedding x is minimal. Conversely, if x is a minimal imbedding, M
has no torsion, and odd-dimensional betti numbers of M vanish, then

G-TOTAL CURVATURE 389
TAn(χ, 1,1) = χ{M)cm_x = Tn(x, 1,1).
By the continuity of Kn(P, e) on the normal bundle Bv and the definitions of
TAn(x,
1,1) and Tn(x, 1,1), we thus obtain Kn(P, e) > 0 everywhere.
(b) Suppose that Kn(P, e) > 0 everywhere, and n > 0. Then from
Kn(P, -e) = (-l)nKn(P, e) it
follows
that dimM = n is even. Let (P, e) be
a
point in Bυ such that the second fundamental form at (P, e) is positive
definite. Then the second fundamental form at (P, —e) is negative definite. By
the
continuity of the second fundamental form on the fibre
S7-'71'1
we see that
if the codimension m — n > 1, then the Lipschitz-Killing curvature Kn(P, e)
=
0 at some points in
S[~~n~ι.
This is impossible by the assumption. Thus we
have m ~ n — 1. In this case, the condition that Kn(P, e) > 0 everywhere
implies that Gauss-Kronecker curvature of M in En+ι is positive everywhere.
Hence
x(M) is a convex hypersurface in En+1.
Remark
7.1. If the codimension m — n = 1, then the sufficiency of
Theorem
7.2, Part (a) was proved by Chern-Lashof [10, II], and Theorem
7.2, Part (b) was the well-known Hadamard theorem. In [10, II], Chern and
Lashof
gave
an example of nonconvex hypersurface in En+1 with Kn(P, e) > 0
everywhere. In [15], Kobayashi
gave
an example of a minimal imbedding of
complex projective spaces in higher dimensional euclidean space; in his
example, the Lipschitz-Killing curvature Kn(P, e) > 0 everywhere.
If C is a closed curve in E3, then we have the so-called curvature k and
torsion
τ. If the torsion τ = 0 identically on C, then C is a plane curve in E\
Moreover, if the curvature k is constant and the torsion τ — 0 identically, then
C
is a circle in a plane of Ez. By using Theorem 7.2 and a result of Chern-
Lashof [10, I], we have
Corollary
7.3. Let x\ M
—>
Em be an
imbedding
of an
even-dimensional
topological
sphere
in Em
with
m — n > 1. Then the
secondary
curvature
μ = 0
when
and
only
when
M is
imbedded
as a convex
hypersurface
in an (n + 1)-
dimensional linear
subspace
of Em.
Moreover,
the
secondary
curvature
μ — 0,
and the
principal
curvature
λ is
constant
when
and
only
when
M is
imbedded
as a
hyper
sphere
in an (n + l)-dimensional linear
subspace
of Em.
8.
Product
immersion
and
immersion
with
constant
G-total
curvature
Proposition
8.1. Let xλ: Mx
—>
Emi and x2: M2 -+ Em2 be
immersions
of
Mx and M2 in Emi and Em<ί
respectively,
and xx x χ2 be the
product
immersion
of xλ and x2. Then
=
Gni(x19
P1? 1,
l)Gn2(x2,
P2, 1, \)cmi+m2_x ,
for all (P19 P2) e Mx x M2,
where
dim Mλ = nx and dim M2 = n2.

390
BANG-YEN
CHEN
This proposition
can be
proved
in the
same
way as
Theorem
10 was
proved
in
[2, I], so we
omit
the
proof.
Corollary
8.2. Let Mx and M2 be two
oriented
closed
manifolds.
Then
the
Euler
characteristics
of Mx and M2
satisfy
(8.2)
χ(Mλ X M2) = χ(Mx) X
χ(M2)
.
This corollary
follows
immediately from Theorem
6.5 and
Proposition
8.1.
Proposition
8.3. Let x: M
—»
Em be an
immersion
of an
oriented
closed
even-dimensional
manifold
M in Em
such
that
x(M) is
contained
in an (n + 1)-
dimensional
linear
subspace
En+1 and the n-th
G-total
curvature
Gn(x, P, 1,1)
>
0
everywhere
on M.
Then
x(M) is a
convex
hyper
sphere
in En+1, and
there
exists
an
oriented
closed
even-dimensional
nonconvex
submanifold
in
En+2
with
positive
constant
n-th
G-total
curvature
Gn(x,P,
1,1).
Proof. The
first
part
follows
from Proposition
8.1 and
Theorem
7.2. Let
S^n
X S^n C
En+2
be the
natural product manifold
of two
unit ^-spheres
in
£w+2
Then this product manifold
in
En+2
has
constant
n-th
G-total curvature
Gn(x,P,
1,1)
everywhere.
By Proposition
8.3 we
have
Corollary
8.4. // M is an
exotic
n-sphere,
then
M
cannot
be
immersed
in
En+1
as a
hypersphere
with
Gn(x, P, 1,1) > 0.
Remark
8.1. Every compact homogeneous space
M can be
immersed
in a
euclidean space with constant
i-th
G-total curvature
Gt(x, P, 1,1).
This immer-
sion
can be
done
by
using equivariant immersion
of M in the
euclidean space.
Remark
8.2.
Let
M be an
n-dimensional manifold immersed
in Em. If i is
an
even positive integer,
2 < i < n,
then
we
have
G,(x,
P, 1,1) =
const.
Σ δl!19 ' "' \
)Rjlhklk2
RJt-lJikt-lkt
,
in
which
RJklh
are the
components
of the
Riemannian-Christoίϊel tensor
(relative
to
orthonormal frames)
of the
induced Riemannian metric
on M, and
δ
11'
""'#*)
does
not
vanish
if and
only
if ]ί9 , jt are
rearrangement
of
\κί9
, Kij
kl9
- 9kt; its
value
is 1 if the
permutation
is
even
and
— 1
if
odd. Hence
we
see that
G^JC,
P, 1,1) are
isometric scalar invariants.
In
fact,
G^JC,
P, 1,1) are
among
the
most important scalar invariants
of the
Riemannian metric.
For
example,
G2(JC,
P, 1,1) =
const.
2
ΛyiiaΛlJfc9
is
called the scalar curvature
of the
Riemannian
metric (see,
for
instance, Chern [7], Nagano [16]).
References
[
1 ] K. Amur,
Vector
forms
and
integral
formulas
for
hypersurfaces
in
Euclidean
space,
J.
Differential
Geometry
3
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G-TOTAL CURVATURE 391
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