with nearly Ricci parallel tensor

擬リッチ平行テンソルを持つケーラー超曲面 Kaehler hypersurfaces

with nearly Ricci parallel tensor

数学専攻   戸田 皓太 TODA Kouta Let ˜M_n+1(˜c) be a complex (n+1)-dimensional complex space form of constant holomorphic sectional curvature ˜c (i.e. complete, simply connected Kaehler manifold with constant holomorphic sectional curvature, say, ˜c). For each real number ˜c, there is (up to holomorphic isometry) exactly one complex space form in every dimension with holomorphic sectional curvature ˜c. If ˜c is positive, then ˜M_n+1(˜c) is the complex projective space P_n+1(C) with the Fubini-Study metric of constant holomorphic sectional curvature ˜c. If ˜c is zero, then ˜M_n+1(˜c) is the complex Euclidean spaceC_n+1. If ˜cis negative, then M˜_n+1(˜c) is the open unit ballD_n+1 inC_n+1 endowed with Bergman metric of constant holomorphic sectional curvature ˜c.

Let M_n be a complex hypersurface in a complex space form ˜M_n+1(˜c). From now on we call such a hypersurface M_n a Kaehler hypersurface. Let ∇and S be the covariant differentiation onM_nand the Ricci tensor ofM_n, respectively.

K. Nomizu and B. Smyth [3] classified these Kaehler hypersurfaces with regard to the parallel Ricci tensor, i.e., ∇S = 0. They proved that if the Ricci tensor S of M_n is parallel, then M_n is locally symmetric, that is, ∇R = 0 and either M_n is totally geodesic in ˜M_n+1(˜c) or M_n is locally the complex quadric in ˜M_n+1(˜c), the latter case arising only when ˜c > 0, where R denotes the curvature tensor of M_n.

The Ricci tensor S is called the neaely Ricci parallel tensor if S satisfies (∇_XS)X = 0

for any X tangent to M_n.

The purpose of this paper is to classify Kaehler hypersurfaces with nealy Ricci parallel tensor in a complex space form. We notice that this condition is weaker than ∇S= 0. We prove the following theorem:

Theorem. Let M_n be a Kaehler hypersurface of the complex dimension n ≥ 2 with nealy Ricci parallel tensor in a complex space form M˜_n+1(˜c). Then

Codazzi equation reduces i.e. ,

(∇_XA)Y −s(X)JAY = (∇_YA)X−s(Y)JAX = 0 for any X and Y tangent to M_n.

Now, let M_n be a Kaehler hypersurface of the complex dimension n in a complex space form ˜M_n+1(˜c) of constant holomorphic sectional curvature ˜c,and

∇˜ (resp. ∇) be the covariant differentiation on ˜M_n+1(˜c) (resp. M_n). For each point x₀ ∈ M_n, we choose an unit normal vector field ξ defined in a neighborhood U(x₀) ofx₀. Denoting the complex structure on ˜M_n+1(˜c) by J, Jξ is also a normal vector field on U(x₀). Then, for any vector fields X, Y tangent to M_n onU(x₀), we have

(1) ∇˜_XY =∇_XY +g(AX, Y)ξ+g(JAX, Y)Jξ, (2) ∇˜_Xξ=−AX+s(X)Jξ,

where g, s are the induced Kaehler metric on M_n, the tensor field of type (0, 1), respectively, and Ais the (1, 1)-type symmetric tensor field and called the second fundamental form. It is easy to show that AJ =−JA.

Let R be the curvature tensor of M_n. Then, for any vector fields X, Y and Z on U(x₀), we have the following:

(3) g(R(X, Y)Z, W)

=g( ˜R(X, Y)Z, W)

+{g(AX, Z)g(AY, W)−g(AX, W)g(AY, Z)} +{g(JAX, Z)g(JAY, W)−g(JAX, W)g(JAY, Z)},

—— Gauss equation where ˜R is the curvature tensor of ˜M_n+1(˜c). Since ˜M_n+1(˜c) is of constant holomorphic sectional curvature ˜c, g( ˜R(X, Y)Z, W) can be written as

(4) g( ˜R(X, Y)Z, W) = c˜

4{g(X, Z)g(Y, W)−g(X, W)g(Y, Z) +g(X, JZ)g(Y, JW)−g(X, JW)g(Y, JZ) + 2g(X, JY)g(Z, JW)}.

Moreover, we denote the (0,2)-type Ricci tensor ofM_n by S. For any pointx of U(x₀),S is defined by

(5) S(X, Y) =

n

i=1

g(R(e_i, X)Y, e_i) +

n

i=1

g(R(Je_i, X)Y, Je_i), where {e₁, . . . , e_n, Je₁, . . . , Je_n} is an orthonormal basis of the tangent space T_xM_n.

Furthermore we need the following results to prove our theorem.

Proposition 1 (Nomizu and Smyth [3]). IfM_nis a Kaehler hypersurface in a complex space form M˜_n+1(˜c), then the following conditions are equivalent on M_n:

(i) Codazzi equation reduces.

(ii) The Ricci tensor of M_n is parallel, that is ∇S = 0.

(iii) M_n is locally symmetric.

Proposition 2 (Smyth[5]). If R˜ and R denote the riemannian curvature tensors of M˜ and M respectively, then for any vector fields X, Y and W on U(x₀), then the ξ-component of R(X, Y˜ )W is given by

g((∇XA)Y −(∇YA)X−s(X)JAY +s(Y)JAX, W)ξ.

Proposition 3 (Smyth[5]). If M˜ is of constant holomorphic sectional cur- vature ˜c, then for any pair of vectors X and Y tangent to M at a point of U(x₀), we have the equations

(6) S(X, Y) =−2g(A²X, Y) + (n+ 1)˜c

2 g(X, Y), (7) (∇XA)Y −s(X)JAY = (∇YA)X−s(Y)JAX.

—— Codazzi equation In particular, if Codazzi equation (7) satisfies

(8) (∇XA)Y =s(X)JAY

on a neighborhood of every point in M_n, then we say that Codazzi equation reduces.

Hence we have

(9) SX = (n+ 1)˜c

2 X−2A²X.

We put

T_λi(x₀) ={X ∈T_x0M_n|AX =λ_iX}. Proposition 4 T_λi and T_−λi are differentiable.

The author would like to express his sincere gratitude to Professor Y. Mat- suyama for his valuable suggestions during the preparation of this paper.

References

[1] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol.

II, Interscience Tracts, John Wiley and Sons, New York, 1963.

[2] Y. Matsuyama, Minimal Submanifolds in S^N and R^N, Mathematische Zeitschrift 175 (1980) 275-282.

[3] K. Nomizu and B. Smyth, Differential geometry of complex hypersurfaces II, J. Math. Soc. Japan Vol. 20, No. 3, 1968.

[4] P. J. Ryan, A class of complex hypersurfaces, Colloquium Mathematicum 26 (1972) 177-182.

[5] B. Smyth,Differential geometry of complex hypersurfaces, Ann. of Math., 85 (1967), 246-266.

Department of Mathematics, Chuo University

1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan