YoshioMatsuyama Onrealhypersurfacesofcomplexprojectivespace

On real hypersurfaces of complex projective space

Yoshio Matsuyama

Abstract

The purpose of the present paper is to survey and to show some new results with respect to characterizations of real hypersurfaces of CP

1 Introduction

Let CP

, n 2, be an n-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4, and let M be a real hypersurface of CP

.Let n be a unit normal vector field on M and x Jn, where J denotes the complex structure of CP

. M has an almost contact metric structure f, x, h, g induced from J.Many differential geometers have studied M cf. 1 , 2 , 3 , 6 , 7 , 13 , 14 , 15 , 18 , 22 and 25 by using the structure f, x, h, g

Typical examples of real hypersurfacces in CP

are homogeneous ones.T 6@6<> 25 showed that all homogeneous real hypersurfaces in CP

are realized as the tubes of constant radius over compact Hermitian symmetric spaces of rank 1 or rank 2.Namely, he showed the following : Let M be a homogeneous real hypersurface of CP

.Then M is a tube of radius r over one of the following Kaehler submanifolds :

A

hyperplane CP

, where 0 r p 2 ,

A

totally geodesic CP

1 k n 2 , where 0 r p 2 , B complex quadric Q

, where 0 r p

4 , C CP

CP

n1

, where 0 r p

4 and n 5 is odd, D complex Grassmann CG

, where 0 r p

4 and n 9, E Hermitian symmetric space SO 10 /U 5 , where 0 r p

4 and n 15.

Due to his classification, we find that the number of distinct constant principal curvatures of a homogeneous real hypersurface is 2, 3 or 5.Here note that the vector x of any homogeneous real hypersurface M which is a tube of radius r is a principal curvature vector with principal curvature a 2 cot 2r with multiplicity 1 See 1 and that in the case of type A

M has two distinct principal curvatures, in the case of type A

resp.B M has three distinct principal curvatures t, 1

t and a t 1

t ῌ ῍

῎ resp. 1 t 1 t , t 1

t 1 and a t 1 t

῏ ῐ

ῑ and in the case of C, D, E, M has five distinct principal curvatures t, 1

t , 1 t 1 t , t 1

t 1 and a t 1

t , respectively.

ῒ

2000 Mathematics Subject Classification : 53B25, 53C40

Keywords and phrases : complex projective space, real hypersurface, geodesic hypersphere, ruled hypersurface

Journal of the Institute of Science and Engineering 5 Chuo University

The purpose of the present paper is to survey with respect to characterizations of real hypersurfaces of CP

and to show the following results :

Theorem 18 A real hypersurface M of CP

satisfies

A Z c g fY, Z fX g fX, Z fY , X, Y and Z x if and only if M is locally congruent to A

.

Theorem 19 A real hypersurface M of CP

satisfies g

A Z, W 0, X, Y, Z and W x if and only if M is locally congruent to A

or A

.

Theorem 20 A real hypersurface M of CP

satisfies

A Z g fY, Z fAX g fAX, Z fY, X, Y and Z x if and only if M is locally congruent to A

or A

.

Theorem 36 A real hypersurface M of CP

satisfies

S Z c g fY, Z fAX g fAX, Z fAY , X, Y, Z x if and only if M is locally congruent to A

, A

and n 2k 1, r p

2 and B and cot

2r n 2.

2 Preliminaries

Let X be a tangent vector field on M. We write JX fX h X n, where fX is the tangent component of JX and h X g X, x As J

Id, where Id denotes the identity endomorphism on TCP

, we get

1 f

X X h X x, h fX 0, fx 0

for any X tangent to M. It is also easy to see that for any X and Y tangent to M

2

f Y h Y AX g AX, Y x,

3

xfAX,

where denotes the convariant differentiation on M. Finally, from the expression of the curvature tensor of CP

, we see that the curvature tensor R, Codazzi equation and the Ricci tensor of M are given by

4 R X, Y Z g Y, Z X g X, Z Y g fY, Z fX g fX, Z fY 2g fX, Y fZ g AY, Z AX g AX, Z AY, 5

A Y

A X h X fY h Y fX 2g fX, Y x.

6 SX 2n 1 X 3h X x hAX A

X.

3 A characterization by the second fundamental form Let M be a real hypersurface of CP

. At first we introduce the following :

Theorem 1 O @JBJG6 22 A real hypersurface M of CP

satisfies Af fA 0 on M if and only if M is locally congruent to A

or A

.

Theorem 2 M 6:96 14 A real hypersurface M of CP

satisfies the equation

A Y h Y fX g fX, Y x

Yoshio Matsuyama

for all X, Y TM if and only if M is locally congruent to A

or A

.

Theorem 3 K >BJG6 9 A real hypersurface M of CP

has constant principal curvatures and the structure vector x which is a principal vector if and only if M is locally congruent to one of homogeneous real hypersurfaces.

Theorem 4 M 6:96 15 There exist no real hypersurfaces which satisfy RA 0 and n 3.

Remark 1 K >BJG6 and M 6:96 12 showed that Theorem 4 remains also true for the case of n 2.

Theorem 5 K >BJG6 and M 6:96 10 A real hypersurface M of CP

satisfies

A 0 if and only if M is locally congruent to A

, A

or a non homogeneous real hypersurface which lies on a tube of radius p

4 over a certain Kaehler submanifold N N in CP

.

Theorem 6 K >BJG6 and M 6:96 12 A real hypersurface M of CP

, n 2 satisfies

R X, Y A Z 0 if and only if M is locally congruent to A

, i.e., a geodesic hypersphere.

Theorem 7 K >BJG6 and M 6:96 12 A real hypersurface M of CP

, n 3 satisfies R X, Y A Z l h X h Z Y h X g Y, Z x h Y h Z X h X g X, Z x if and only if M is locally congruent to a geodesic hypersphere.

Theorem 8 G DID= 2 A real hypersurface M of CP

, n 3 satisfies R X, Y A Z 0, X, Y and Z x

if and only if M is locally congruent to a geodesic hypersphere.

Next, we introduce with respect to recurrent hypersurfaces :

Theorem 9 H 6B696 3 There exist no real hypersurfaces which satisfy

A Y a X AY for a 1 -form a we call a recurrent hypersurface such a hapersurface and have the structure vectors x being principal vectors.

Theorem 10 H 6B696 3 A real hypersurface M of CP

satisfies g

A Y, Z a X g AY, Z , X, Y and Z x

we call an h-recurrent hypersurface such a hypersurface and has the sturucture vector x which is a principal vector if and only if M is locally congruent to A

, A

or B.

Theorem 11 S J= 24 An h-recurrent real hypersurface M of CP

satisfies g Af fA X, Y 0, X, Y x

if and only if M is locally congruent to A

, A

or a ruled real hypersurface.

Moreover, we introduce with respect to birecurrent hypersurfaces :

Theorem 12 N 6@6?>B6 20 There exist no real hypersurfaces which satisfy

A Z

A Z

A Z a X, Y AZ for a 2-form a we call a birecurrent hypersurface such a hypersurface

Theorem 13 N 6@6?>B6 20 A real hypersurface M of CP

satisfies g

A Z,W g fAX, W

g fY, Z g fAX, Z g fY, W , X, Y, Z and W x

and has the structure vector x which is a principal

vector if and only if M is locally congruent to A

or A

.

Finally of this section, we borrow from the way of G DID= 2 :

Theorem 14 M 6IHJN6B6 16 A real hypersurface M of CP

satisfies R AX, Y Z A R X, Y Z 0, X, Y and Z TM if and only if M is locally congruent to A

or A

.

Theorem 15 M 6IHJN6B6 17 A real hypersurface M of CP

satisfies R AX, Y Z A R X, Y Z 0, X, Y and Z x

if and only if M is locally congruent to A

or A

.

Theorem 16 M 6IHJN6B6 18 A real hypersurface M of CP

satisfies g R AX, Y Z A R X, Y Z , W 0, X, Y, Z and W x

if and only if M is locally congruent to A

, A

or a ruled real hypersurface.

Theorem 17 N 6@6?>B6 21 A real hypersurface M of CP

satisfies g R X, Y A Z, W 0, X, Y, Z and W x

if and only if M is locally congruent to A

or a ruled real hypersurface.

Hence we obtain the following:

Theorem 18 A real hypersurface M of CP

satisfies

A Z c g fY, Z fX g fX, Z fY , X, Y and Z x

if and only if M is locally congruent to A

.

Proof. From the assumption we have R X, Y A Z 0, X, Y and Z x

. By Theorem 8 we obtain the conclusion.

Theorem 19 A real hypersurface M of CP

satisfies g

A Z, W g fAX, W g fY, Z g fAX, Z g fY, W , X, Y, Z and W x

if and only if M is locally congruent to A

or A

.

Proof. In terms of the assumption we get g R X, Y A Z, W 0, X, Y, Z and W x

. From Theorem 17 we have the result.

Similarly, we obtain :

Theorem 20 A real hypersurface M of CP

satisfies

A Z g fY, Z fAX g fAX, Z fY, X, Y and Z x

if and only if M is locally congruent to A

or A

.

4 A characterization by Ricci tensor

Let M be a real hypersurface of CP

. First of all we introduce the following:

Theorem 21 C :8>A , R N6C 1 and K DC 13 A real hypersurface M of CP

satisfies SX aX bh X x for some functions a and b we call an h-Einstein such a real hypersurface if and only if M is locally congruent to A

, A

and cot

r k

n k 1 or B and cot

2r n 2.

Theorem 22 K >BJG6 8 A real hypersurface M of CP

satisfies the equation

X

S Y c g fAX, Y x h Y fAX , c nonzero const.

for all X, Y TM if and only if M is locally congruent to A

, A

or B.

Yoshio Matsuyama

Theorem 23 K > , N 6@6<6L6 and S J= 6 A real hypersurface M of CP

satisfies

R X, Y S Z 0 if and only if M is an h Einstein.

Theorem 24 K > , N 6@6<6L6 and S J= 6 There exist no real hypersurfaces which satisfy RS 0.

Theorem 25 S J= 23 A real hypersurface M of CP

satisfies g

S Y, Z 0, X, Y and Z x

and has the structure vector x which is a principal vector if and only if M is locally congruent to A

, A

or B.

Theorem 26 K >BJG6 and M 6:96 10 A real hypersurface M of CP

satisfies

S 0 and has constant mean curvature and the structure vector x which is a principal vector has the correspondingprincipal curvature beingnonzero if and only if M is locally congruent to A

and r p

4 , A

and r p

4 , B and cot

2r n 2, C, D and E.

Theorem 27 K >BJG6 and M 6:96 11 A real hypersurface M of CP

satisfies

S Y c g fX, Y x h Y fX , c const. if and only if M is locally congruent to a geodesic hypersphere.

Theorem 28 K >BJG6 and M 6:96 12 A real hypersurface M of CP

satisfies

S Y l g fX, Y xh Y fX if and only if M is locally congruent to a geodesic hypersphere.

Theorem 29 K >BJG6 and M 6:96 12 A real hypersurface M of CP

satisfies R X, Y S Z m h Y g X, Z x h Z X h X g Y, Z x h Z Y if and only if M is locally congruent to A

or A

and n 2 k 1, r p

4 .

Second, we introduce with respect to recurrent hypersurfaces :

Theorem 30 H 6B696 4 There exist no real hypersurfaces which satisfy

S Y a X SY for a 1 -form a we call a Ricci recurrent hypersurface such a hypersurface and have the structure vectors x being principal vectors.

In addition, we introduce with respect to birecurrent hypersurfaces.

Theorem 31 I @JI6 5 There exist no real hypersurfaces which satisfy

S Z

S Z

xY

S Z a X, Y SZ for a 2-form a we call a Ricci birecurrent hypersurface such a hypersurface

Theorem 32 I @JI6 5 There exist no real hypersurfaces which satisfy g

S Z, W a X, Y g SZ, W , X, Y, Z and W x

we call a Ricci h birecurrent hypersurface such a hypersurface and have the structure vectors x beingprincipal vectors.

Theorem 33 I @JI6 5 A real hypersurface M of CP

satisfies g

S Z, W c g fAX, W g fAY, Z g fAX, Z g fY, W , X, Y, Z and W x

and has the sturucture vector which is a principal vector if and only if M is locally congruent to A

and c 2 ; A

, cot

r k

n k 1 and c 2 and B, cot

2r n 2 and c 2 2n 1

Lastly, we borrow from the way of G DID= 2 :

Theorem 34 M 6IHJN6B6 19 A real hypersurface M of CP

satisfies

R X, Y S Z 0 if and

only if M is h Einstein.

Theorem 35 M 6IHJN6B6 19 A real hypersurface M of CP

satisfies R X, Y S Z 0, X, Y and Z x

if and only if M is locally congruent to A

, A

and n 2k 1, r p

4 and B and cot

2r n 2.

Hence we obtain the following :

Theorem 36 A real hypersurface M of CP

satisfies

S Z c g fY, Z fAX g fAX, Z fAY , X, Y, Z x

if and only if M is locally congruent to A

, A

and n 2k 1, r p

4 and B and cot

2r n 2.

References

1 C :8>A , T. 6C9 R N6C , P.J., Focal sets and real hypersurfaces in complex projective space, Trans. Amer.

Math. Soc. 269, pp.481-498 1982

2 G DID= , T., Geodesic hypersurfaces in complex projective space, Tsukuba J. Math. 18, pp.207-215 1994

3 H 6B696 , T., On real hypersurfaces of a complex projective space with h-recurrent second fundamental tensor, Nihonkai Math. J. 6, pp.153-163 1995

4 H 6B696 , T., On real hypersurfaces of a complex projective space with recurrent Ricci tensor, Glasgow Math. J. 41, pp.297-302 1999

5 I @JI6 , K., On real hypersurfaces of a complex projective space, J. Korean Math. Soc. 6, pp.153-163 1999

6 K > , U-H., N 6@6<6L6 , H. 6C9 S J= , Y.J. Real hypersurfaces with harmonic Weyl tensor of a complex space form, Hiroshima Math. J. 20, pp.93-102 1990

7 K >BJG6 , M., Real hypersurfaces and complex submanifolds in complex projective space Trans. Amer.

Math. Soc. 296, pp.137-149 1986

8 K >BJG6 , M., Real hypersurfaces of a complex projective space Bull. Austral. Math. Soc. 33, pp.383-387 1986

9 K >BJG6 , M., Sectional curvatures of holomorphic plances on a real hypersurfaces in P

ῌ , Math.

Ann. 276, pp.487-497 1987

10 K >BJG6 , M. and M 6:96 , S, On real hypersurfaces of a complex projective space ῌ , Tsukuba J. Math.

15, pp.547-561 1991

11 K >BJG6 , M. and M 6:96 , S, Characterization of geodesic hypersurfaces in a complex projective space in terms of Ricci tensors, Yokohama Math. J. 40, pp.35-43 1992

12 K >BJG6 , M. and M 6:96 , S, On real hypersurfaces of a complex projective space ῍ , Hokkaido Math. J.

22, pp.63-78 1993

13 K DC , M., Pseudo-Einstein real hypersurfaces in complex space forms, J. Differential Geom. 14, pp.339- 354 1979

14 M 6:96 , Y., On real hypersurfaces in a complex projective space, J. Math. Soc. Japan 26, pp.529-540 1976

15 M 6:96 , S., Real hypersurfaces of complex projective spaces, Math. Ann. 263, pp.473-478 1983 16 M 6IHJN6B6 , Y., A characterization of real hypersurfaces in complex projective space, J. Institute Sci.

and Eng. Chuo Univ. 2, pp-11-13 1996

17 M 6IHJN6B6 , Y., A characterization of real hypersurfaces in complex projective space ῌ , J. Institute Sci. and Eng. Chuo Univ. 3, pp.1-3 1997

18 M 6IHJN6B6 , Y., A characterization of real hypersurfaces in complex projective space ῍, Yokohama

Yoshio Matsuyama

Math. J. 46, pp.119-126 ῍ 1999 ῎ῌ

῏ 19 ῐ M 6IHJN6B6 , Y., Real hypersurfaces in complex prejective space satisfying a certain condition on Ricci tensors, Nihonkai Math. J. 10, pp.171-178 ῍ 1999 ῎ῌ

῏ 20 ῐ N 6@6?>B6 . S., Real hypersurfaces of a complex projective space, Tsukuba J. Math. 28, pp.119-126

῍ 1999 ῎ῌ

῏ 21 ῐ N 6@6?>B6 . S., On real hypersurfaces of a complex projective space, preprint.

῏ 22 ῐ O @JBJG6 , M., On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212, pp.355-364 ῍ 1975 ῎ῌ

῏ 23 ῐ S J= , Y. J., On real hypersurfaces of a complex space form with h-parallel Ricci tensor, Tsukuba J. Math.

14, pp.27-37 ῍ 1990 ῎ῌ

῏ 24 ῐ S J= , Y.J., Real hypersurfaces in complex space forms with h-recurrent second fundamental tensors, Math. J. Toyama. Univ. 19, pp.127-141 ῍ 1996 ῎ῌ

῏ 25 ῐ T 6@6<> , R., On real hypersurfaces of a complex projective space, Osaka J. Math. 10, pp.495-506

῍ 1973 ῎ῌ

Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, JAPAN

e-mail address matuyama ῒ math.chuo-u.ac.jp