On normal connection of Kaehler submanifolds

Abstract

In this paper, we shall study the normal connection of a Kaehler submanifold $M$ in another Kaehler manifold $\tilde{M}$. In section 3, we shall prove that the normal connection of $M$ in $\tilde{M}$ is flat only when the Ricci tensors of $M$ and $\tilde{M}$ are equal on the tangent bundle of $M$. Moreover, we shall prove that if $M$ and $\tilde{M}^{m}$ are both compact and $\tilde{M}$ is flat then the normal connection is flat when and only when the first Chern class $c_{1}(\nu)$ of the normal bundle $\nu$ is trivial. In section 4, we shall prove that the complex projective line in a complex sphere $Q_{n}=$ $SO(n+2)/SO(2)XSO(n)$ is the only Kaehler submanifold of $Q_{n}$ whose normal bundle admits a parallel section. Moreover, the complex projective line in $Q_{2}$ is the only Kaehler submanifold in $Q_{n}$ with flat normal connection.

Full text

J.
Math.
Soc.
Japan
Vol.
27,
No.
4,
1975
On
normal
connection
of
Kaehler
submanifolds
By
Bang-yen
CHENl)
and
Huei-shyong
LUE
(Received
Nov.
8,
1974)
\S 1.
Introduction.
Let
$M$
be
an
n-dimensional
Riemannian
manifold
with
Levi-Civita
connec-
tion
$\nabla$
.
Then
the
curvature
tensor
$R$
of
$M$
is
given
by
$R(X, Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}$
$-\nabla_{[X,Y]}$
for
any
tangent
vector
fields
$X$
and
$Y$
.
Let
$E_{1},$
$\cdots$
,
$E_{n}$
be
an
ortho-
normal
frame
on
$M$
.
Then
the
Ricci
tensor
$S(X, Y)$
and
the
scalar
curvature
$\rho$
are
given
respectively
by
$S(X, Y)=\sum_{t=1}^{n}R(E_{i}, X ; Y, E_{i})$
,
$\rho=\frac{1}{n}\sum_{i=1}^{n}S(E_{i}, E_{i})$
,
where
$R(E_{i}, X;Y, E_{i})=g(R(E_{i}, X)Y,$
$E_{i}$
)
and
$g$
is
the
metric
tensor
of
$M$
.
Let
$x:M\rightarrow\tilde{M}$
be
an
isometric
immersion
of
$M$
into
an
$m$
-dimensional
Rie-
mannian
manifold
$\tilde{M}^{m}$
with
connection
$\tilde{\nabla}$
and
metric
tensor
$\tilde{g}$
.
Then
the
second
fundamental
form
$h$
of
$M$
in
$\tilde{M}$
is
given
by
$\tilde{\nabla}_{X}Y=\nabla_{X}Y+h(X, Y)$
.
Let
$N$
be
a
normal
vector
field
of
$M$
in
$\tilde{M}$
,
we
write
$\tilde{\nabla}_{X}N=-A_{N}(X)+D_{X}N$
,
where
$-A_{N}(X)$
and
$D_{X}N$
denote
the
tangential
and
normal
components
of
$\tilde{\nabla}_{X}N$
.
Then
we
have
$g(A_{N}(X), Y)=\tilde{g}(h(X, Y),$
$N$
).
$D$
is
called
the
normal
connection
of
$M$
in
$\tilde{M}^{m}$
.
A
local
normal
vector
field
$N\neq 0$
is
called
a
parallel
section
if
$DN=0$
.
Let
$R^{\perp}$
be
the
curvature
tensor
associated
with
$D$
,
$i$
.
$e.$
,
$R^{\perp}(X, Y)=D_{X}D_{Y}-D_{Y}D_{X}-D_{[X,Y]}$
.
Then
the
normal
connection
$D$
is
flat
if
$R^{\perp}$
vanishes
identically.
The
normal
connection
is
flat
if
the
(real)
codimension
is
one.
If
the
(real)
codimension
is
higher,
then
the
normal
connection
is
not
flat
in
general.
In
this
paper,
we
shall
study
the
normal
connection
of
a
Kaehler
submani-
fold
$M$
in
another
Kaehler
manifold
$\tilde{M}$
.
In
\S 3,
we
shall
prove
that
the
normal
connection
of
$M$
in
$\tilde{M}$
is
flat
only
when
the
Ricci
tensors
of
$M$
and
$\tilde{M}$
are
equal
on
the
tangent
bundle
of
$M$
.
Moreover,
we
shall
prove
that
if
$M$
and
$\tilde{M}^{m}$
are
both
compact
and
$\tilde{M}$
is
flat
then
the
normal
connection
is
flat
when
and
only
when
the
first
Chern
class
$c_{1}(\nu)$
of
the
normal
bundle
$\nu$
is
trivial.
In
1)
Work
done
under
partial
support
by
NSF
Grant
GP-36684.

Normal
connection
of
Kaehler
submanifolds
551
\S 4,
we
shall
prove
that
the
complex
projective
line
in
a
complex
sphere
$Q_{n}=$
$SO(n+2)/SO(2)XSO(n)$
is
the
only
Kaehler
submanifold
of
$Q_{n}$
whose
normal
bundle
admits
a
parallel
section.
Moreover,
the
complex
projective
line
in
$Q_{2}$
is
the
only
Kaehler
submanifold
in
$Q_{n}$
with
flat
normal
connection.
\S 2.
Basic
formulas.
Let
$M^{n}$
be
a
complex
n-dimensional
Kaehler
manifold
with
complex
struc-
ture
$J$
and
metric
tensor
$g$
.
Then
the
curvature
tensor
$R$
of
$M^{n}$
satisfies
the
following
formulas.
(2.1)
$R(JX, JY)=R(X, Y)$
,
$R(X, Y)JZ=JR(X, Y)Z$
(2.2)
$R(X, Y)Z+R(Y, Z)X+R(Z, X)Y=0$
(2.3)
$R(X, Y;Z, W)=R(Z, W;X, Y)=-R(Y, X;Z, W)$
$=-R(X, Y;W, Z)$
.
Let
$M^{n}$
be
isometrically
immersed
in
a
complex
m-dimensional
Kaehler
mani-
fold
$\tilde{M}^{m}$
as
a
complex
submanifold.
Let
$\tilde{J},\tilde{R}$
and
$\tilde{g}$
be
the
complex
structure,
the
curvature
tensor
and
the
metric
tensor
of
$\tilde{M}^{m}$
,
respectively.
Then
the
equations
of
Gauss
and
Ricci
are
given
respectively
by
(2.4)
$\tilde{R}(X, Y;Z, W)=R(X, Y;Z, W)+\tilde{g}(h(X, Z),$
$h(Y, W))$
$-\tilde{g}(h(Y, Z),$
$h(X, W))$
,
(2.5)
$\tilde{R}(X, Y ; N, N^{\prime})=R^{\perp}(X, Y ; N, N^{\prime})-g([A_{N}, A_{N^{\prime}}](X), Y)$
,
where
$X,$
$Y,$
$Z,$
$W$
are
vector
fields
tangent
to
$M^{n}$
and
$N,$
$N^{\prime}$
are
vector
fields
normal
to
$M^{n}$
.
Moreover,
we
have
(2.6)
$A_{JN}^{\sim}=JA_{N}$
and
$JA_{N}=-A_{N}J$
,
from
which
we
have
trace
$h=0$
.
\S 3.
Ricci
tensor
and
normal
connection.
Let
$M^{n}$
be
a
Kaehler
submanifold
in
another
Kaehler
manifold
$\tilde{M}^{m}$
as
in
\S 2.
Suppose
$N$
be
a
parallel
section
in
normal
bundle
$\nu$
.
Then
$R^{\perp}(X, Y)N=0$
for
all
vector
fields
$X,$
$Y$
tangent
to
$M^{n}$
.
From
the
equation
of
Ricci,
we
find
(3.1)
$\tilde{R}(X, Y;N,\tilde{J}N)=-g([A_{N}, A_{JN}^{\sim}](X), Y)$
.
Hence,
by
using
(2.6),
we
have
(3.2)
$R(X, Y;N, JN)=2g(JA_{N}(X), Y)$
.

552
B.-y.
CHEN
and
H.-s.
LUE
Let
$H_{B}(X, N)$
denote
the
holomorphic
bisectional
curvature
for
the
pair
$(X, N)$
.
Then
we
have
$H_{B}(X, N)=R(X, JX;JN, N)/g(X, X)g(N, N)$
.
From
(3.2)
we
have
the
following
Proposition.
PROPOSITION
1.
Let
$M^{n}$
be
a
Kaehler
submanifold
of
a
Kaehler
manifold
$\tilde{M}^{m}$
.
If
there
is
a
unit
tangent
vector
$X$
such
that,
for
all
unit
normal
vectors
$N$
,
the
holomorPhic
bisectional
curvatures
$H_{B}(X, N)$
are
positive,
then
the
normal
bundle
admits
no
Parallel
section.
In
[5]
Smyth
proved
that
the
normal
connection
of
a
Kaehler
hypersurface
$M^{n}$
in
$\tilde{M}^{n+1}$
is
flat
if
and
only
if
$S(X, Y)=S(X, Y)$
for
all
$X,$
$Y$
in
$TM^{n}$
.
In
this
section
we
shall
prove
the
following.
THEOREM
2.
Let
$M^{n}$
be
a
Kaehler
submanifold
of
a
Kaehler
manifold
$\tilde{M}^{m}$
.
If
the
normal
connection
of
$M^{n}$
in
$\tilde{M}^{m}$
is
flat,
then
the
Ricci
tensors
$S$
and
$\tilde{S}$
of
$M^{n}$
and
$\tilde{M}^{m}$
satisfy
the
following
relation:
$S(X, Y)=S(X, Y)$
for
all
$X,$
$Y$
$\in TM^{n},$
$TM^{n}$
being
the
tangent
bundle
of
$M^{n}$
.
PROOF.
Let
$M^{n}$
be
an
n-dimensional
Kaehler
submanifold
of
an
m-dimen-
sional
Kaehler
manifold
$\tilde{M}^{m}$
with
flat
normal
connection.
Then,
by
Proposition
1.1
in
[1,
p.
99],
there
exist
locally
$2m-2n$
mutually
orthogonal
unit
normal
vector
Pelds
$N_{1},$
$N_{2},$
$\cdots$
,
$N_{2m-2n}$
such
that
$DN_{r}=0$
for
all
$r=1,2,$
$\cdots$
,
$2m-2n$
.
Since
$\tilde{M}^{m}$
is
Kaehlerian,
$\nabla J=0$
,
we
see
that
$N_{1},$
$N_{2},$
$\cdots$
,
$N_{m-n},\tilde{J}N_{1},$
$JN_{m-n}$
are
orthonormal
parallel
sections
in
the
normal
bundle.
From
the
definition
of
Ricci
tensors
and
the
equation
of
Gauss,
we
have
(3.3)
$S(X, Y)=\tilde{S}(X, Y)-\sum_{\alpha=1}^{m-n}\{R(N_{\alpha}, X;Y, N_{\alpha})+\tilde{R}(\tilde{J}N_{\alpha}, X;Y,\tilde{J}N_{a})\}$
$-\sum_{A=1}^{2n}\tilde{g}(h(E_{A}, X),$
$h(E_{A}, Y))$
,
where
$E_{1},$
$\cdots$
,
$E_{2n}$
is
an
orthonormal
frame
of
$M^{n}$
.
On
the
other
hand,
since
$N_{a},$
$\alpha=1,$
$\cdots$
,
$m-n$
are
parallel,
(3.2)
implies
(3.4)
$fi(X, Y, N_{\alpha}, JN_{\alpha})=2g(JA_{N_{\alpha}}^{2}(X), Y)$
.
By
(2.2)
and
(2.3),
we
have
(3.5)
$R(X, JY;N_{\alpha}, JN_{a})=R(N_{\alpha}, JY;X, JN_{\alpha})-R(N_{a}, X;JY,\tilde{J}N_{\alpha})$
.
Hence,
by
using
(2.1)
and
(2.3),
we
have
(3.6)
$R(X, JY;N_{a},\tilde{J}N_{a})=-[R(\tilde{J}N_{a}, X;Y,\check{J}N_{\alpha})+\tilde{R}(N_{a}, X;\prime Y, N_{\alpha})]$
.
Moreover,
from
(2.6),
we
may
find

Normal
connection
of
Kaehler
submanifolds
553
(3.7)
$\sum_{A=1}^{2n}\tilde{g}(h(E_{A}, X),$
$h(E_{A}, Y))=2\sum_{\alpha=1}^{m-n}g(A_{\alpha}^{2}(X), Y)$
,
where
$A_{\alpha}=A_{N_{\alpha}}$
.
Combining
(3.3),
(3.4),
(3.6)
and
(3.7),
we
find
$S(X, Y)=S(X, Y)$
for
all
vector
fields
$X,$
$Y$
tangent
to
$M^{n}$
.
This
completes
the
proof.
A
Kaehler
manifold
$M^{n}$
is
called
an
Einstein
space
if
there
exists
a
func-
tion
$\rho$
on
$M^{n}$
such
that
$S(X, Y)=\rho g(X, Y)$
for
all
tangent
vectors
$X$
and
$Y$
.
The
function
$\rho$
is
the
scalar
curvature
of
$M^{n}$
.
If
$n>1,$
$\rho$
is
constant.
A
Kaehler
manifold
$M^{n}$
is
called
a
complex
sPace
form
of
holomorphic
curvature
$c$
if
the
curvature
tensor
$R$
satisfies
(3.8)
$R(X, Y)Z=\frac{c}{4}\{g(Y, Z)X-g(X, Z)Y+g(JY, Z)JX$
$-g(JX, Z)JY+2g(X, JY)JZ\}$
.
From
Theorem
2,
we
have
immediately
the
following
THEOREM
3.
Let
$M^{n}$
be
a
Kaehler
submanifold
of
a
Kaehler-Einstein
mani-
fold
$\tilde{M}^{m}$
.
If
the
nomal
connection
is
flat,
then
$M^{n}$
is
also
Einstein.
Moreover,
$M^{n}$
and
$\tilde{M}^{m}$
have
the
same
scalar
curvature.
Let
$M^{n}$
and
$\tilde{M}^{m}$
be
both
compact.
If
$m>n+1$
,
then
$S(X, Y)=S(X, Y)$
for
all
$X,$
$Y\in TM^{n}$
seems
to
be
too
weak
to
conclude
the
flatness
of
the
normal
connection.
However
we
have
the
following.
THEOREM
4.
Let
$M^{n}$
be
a
$comPact$
Kaehler
submanifold
of
a
$comPact$
Kaehler
manifold
$\tilde{M}^{m}$
.
Then
we
have
(a)
$S(X, Y)=S(X, Y)$
for
all
$X,$
$Y\in TM^{n}$
implies
$c_{1}(\nu)=0$
,
where
$c_{1}(\nu)$
denotes
the
first
Chern
class
of
the
normal
bundle
$\nu$
.
(b)
If
$\tilde{M}^{m}$
is
flat,
then
the
normal
connection
is
flat
if
and
only
if
$c_{1}(\nu)$
is
zero.
PROOF.
Let
$\Phi$
be
the
fundamental
2-form
on
$M^{n},$
$i$
.
$e.$
,
a
closed
2-form
defined
by
$\Phi(X, Y)=\frac{1}{2}g(JX, Y)$
.
Let
$\tilde{\gamma}$
(respectively,
$\gamma$
)
be
the
Ricci
2-form
of
$\tilde{M}^{m}$
(respectively,
$M^{n}$
)
$i$
.
$e.$
,
a
closed
2-form
defined
by
(3.9)
$\tilde{\gamma}(\tilde{X},\tilde{Y})=\frac{1}{4\pi}\tilde{S}(\tilde{J}\tilde{X},\tilde{Y})(respectively,$
$\gamma(X, Y)=\frac{1}{4\pi}S(JX, Y))$
.
Then
the
first
Chern
class
$c_{1}(T\tilde{M}^{m})$
of
$T\tilde{M}^{m}$
is
represented
by
$\tilde{\gamma}$
(respectively,
$c_{1}(TM^{n})$
of
$TM^{n}$
is
represented
by
$\gamma$
).
Now
suppose
that
$S=\tilde{S}$
on
$TM^{n}$
,
then,
equation
(3.9)
implies
$\tilde{\gamma}|_{M^{n}}=\gamma$
.
Hence
we
have
(3.10)
$c_{1}(T\tilde{M}^{m}|_{M^{n}})=c_{1}(TM^{n})$
.

554
B.-y.
CHEN
and
H.-s.
LUE
On
the
other
hand,
since
$ T\tilde{M}^{m}|_{M^{n}}=TM^{n}\oplus\nu$
,
we
find
(3.11)
$c_{1}(T\tilde{M}^{m}|_{M^{n}})=c_{1}(TM^{n})+c_{1}(\nu)$
.
Substituting
(3.10)
into
(3.11),
we
get
$c_{1}(\nu)=0$
.
This
proves
(a).
Now,
suppose
that
$\tilde{M}^{m}$
is
flat
and
$c_{1}(\nu)=0$
.
Then,
by
(3.9)
and
(3.11),
we
have
$c_{1}(TM^{n})=0$
.
Hence,
there
exists
a
l-form
$\eta$
such
that
(3.12)
$\gamma=d\eta$
.
Let
$\Lambda$
be
the
operator
of
interior
product
by
$\Phi$
.
Applying
$\Lambda$
to
both
sides
of
(3.12)
we
have
(3.13)
$ n\rho=4\pi\Lambda d\eta$
.
Let
$\delta$
be
the
codifferential
operator
and
$C$
the
operator
defined
by
$C\alpha=$
$(\sqrt{-1})^{r-s}\alpha$
,
where
$\alpha$
is
a
form
of
type
$(r, s)$
.
Then
by
using
the
well-known
identity
$ d\Lambda-\Lambda d=\delta C-C\delta$
,
we
have
$\Lambda d\eta=-\delta C\eta$
since
$d\Lambda\eta=C\delta\eta=0$
.
Thus
we
find
(3.14)
$\int_{M^{n}}\rho*1=0$
.
On
the
other
hand,
the
flatness
of
$\tilde{M}^{m}$
and
the
equation
(3.3)
imply
$n\rho=-\Vert h\Vert^{2}$
where
$\Vert h\Vert$
is
the
length
of
$h$
.
Hence,
by
using
(3.14),
we
find
$\rho=h=0$
,
from
which
we
find
$R^{\perp}=0$
.
The
remaining
part
of
this
theorem
is
trivial.
This
proves
the
theorem.
\S 4.
Kaehler
submanifold
in
$Q_{n}$
with
parallel
normal
sections.
Let
$P_{m+1}(c)$
be
an
$(m+1)$
-dimensional
complex
projective
space
with
holo-
morphic
sectional
curvature
4.
Let
$z_{0},$
$z_{1},$
$\cdots$
,
$z_{m+1}$
be
homogeneous
coordinates
in
$P_{m+1}(c)$
.
Then
the
complex
sphere
$Q_{m}$
is
a
complex
hypersurface
of
$P_{m+1}(c)$
dePned
by
the
equation
$z_{0}^{2}+z_{1}^{2}+$
$\cdot$
..
$+z_{m+1}^{2}=0$
.
It
is
well-known
that
the
Hermitian
symmetric
space
$SO(m+2)/SO(2)\times SO(m)$
is
complex
analytically
isometric
to
the
complex
sphere
$Q_{m}$
.
THEOREM
5.
Let
$M^{n}$
be
an
n-dimensional
Kaehler
submanifold
of
$Q_{m}$
.
(a)
If
the
normal
bundle
of
$M^{n}$
in
$Q_{m}$
admits
a
parallel
section,
then
$n=1$
,
$i$
.
$e.,$
$M^{n}$
is
a
holomorphic
curve
in
$Q_{m}$
.
(b)
If
the
normal
connection
of
$M^{n}$
in
$Q_{m}$
is
flat,
then
$n=1$
and
$m=2$
.
Moreover,
$M^{1}$
is
a
linear
curve
in
$P_{3}(c)$
.

Normal
connection
of
Kaehler
submanifolds
555
PROOF.
(a)
Let
$N$
be
a
parallel
section
in
the
normal
bundle.
Then,
for
any
vector
$X$
tangent
to
$M^{n}$
,
equation
(3.2)
implies
that
(4.1)
$R(X, JX;N, JN)=2g(A_{N}(X), A_{N}(Y))$
.
On
the
other
hand,
let
$\tilde{A}$
be
the
operator
associated
with
the
second
funda-
mental
form
of
the
immersion
of
$Q_{m}$
into
$P_{m+1}(c)$
.
Then
(3.8)
and
the
equation
of
Gauss
imply
that
(4.2)
$\tilde{R}(X, JX;N,\tilde{J}N)=2\{\tilde{g}(X,\tilde{A}(N))^{2}+\tilde{g}(JX,\tilde{A}(N))^{2}\}$
$-2g(X, X)\tilde{g}(N, N)$
.
Hence
from
(4.1)
and
(4.2)
we
get
(4.3)
$\tilde{g}(X,\tilde{A}(N))^{2}+\tilde{g}(JX,\tilde{A}(N))^{2}=g(X, X)\tilde{g}(N, N)+g(A_{N}(X), A_{N}(X))$
.
Since
$N$
has
nonzero
constant
length,
(4.3)
implies
that
$\tilde{g}(X,\tilde{A}(N))^{2}+\tilde{g}(JX,\tilde{A}(N))^{2}\neq 0$
for
any
nonzero
vector
$X$
tangent
to
$M^{n}$
.
This
is
clearly
impossible
if
$n\geqq 2$
.
(b)
If
the
normal
bundle
of
$M^{n}$
in
$Q_{m}$
is
flat,
then
there
exists
$2m-2n$
local
parallel
sections.
Hence,
from
part
(a),
we
see
that
$n=1$
.
On
the
other
hand,
from
Theorem
2,
we
have
(4.4)
$S(X, X)=\tilde{S}(X, X)$
for
all
vector
$X$
tangent
to
$M^{1}$
.
Since
$Q_{m}$
is
Einstein
with
$\tilde{S}(X, X)=2mg(X, X)$
.
Hence,
$M^{1}$
is
of
constant
holomorphic
sectional
curvature
$2m$
.
On
the
other
hand,
if
we
regard
$Q_{m}$
as
a
hypersurface
in
$P_{m+1}(C)$
,
then,
by
the
equation
of
Gauss,
we
find
that
$m=2$
,
and
$M^{1}$
is
a
linear
curve
in
$P_{3}(C)$
.
REMARK
1.
$Q_{2}$
is
complex
analytically
isometric
to
$P_{1}(C)\times P_{1}(C)$
.
Hence,
if
we
regard
$P_{1}(C)$
as
a
Kaehler
submanifold
of
$Q_{2}$
in
a
natural
way,
then
the
normal
connection
of
$P_{1}(C)$
in
$Q_{2}$
is
flat.
Let
$Q_{2}$
be
imbedded
in
$Q_{m}$
as
a
totally
geodesic
submanifold
$(m>2)$
.
Then
the
normal
bundle
of
$P_{1}(C)$
in
$Q_{m}$
admits
a
parallel
section.
REMARK
2.
The
normal
bundle
of
Kaehler
submanifolds
in
a
complex
space
form
of
holomorphic
sectional
curvature
$c\neq 0$
admits
no
parallel
section
(Chen-Ogiue
[2]).
(For
hypersurface
case,
see
Kon
[3],
Nomizu-Smyth
[4]
and
Smyth
[5].)
References
[1]
B.-Y.
Chen,
Geometry
of
submanifolds,
Marcel
Dekker,
New
York,
1973.
[2]
B.-Y.
Chen
and
K.
Ogiue,
Some
extrinsic
results
for
Kaehler
submanifolds,
Tamkang
J.
Math.,
4
(1973),
207-213.

556
B.-y.
CHEN
and
H.-s.
LUE
[3]
M.
Kon,
Kaehler
immersions
with
trivial
normal
connection,
TRU
Math.,
9
(1973),
29-33.
[4]
K.
Nomizu
and
B.
Smyth,
Differential
geometry
of
complex
hypersurface,
II,
J.
Math.
Soc.
Japan,
20
(1968),
498-521.
[5]
B.
Smyth,
Differential
geometry
of
complex
hypersurfaces,
Ann.
of
Math.,
85
(1967),
246-266.
Bang-yen
CHEN
Department
of
Mathematics
Michigan
State
University
East
Lansing,
Michigan
48824
U.
S.
A.
Huei-shyong
LUE
Department
of
Mathematics
Michigan
State
University
East
Lansing,
Michigan
48824
U.
S.
A.
Current
address:
Department
of
Mathematics
National
Tsing
$\cdot$
Hua
University
Hsinchu,
Taiwan