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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