Shizuno SEKIGUCHI Let ˜ M n+1 (c) be an (n + 1)-dimensional real space form of constant sectional curvature c.

周期的リッチ平行テンソルを持つ超曲面について On hypersurfaces with cyclic Ricci parallel tensor

数学専攻 関口静乃

Shizuno SEKIGUCHI Let ˜ M ⁿ⁺¹ (c) be an (n + 1)-dimensional real space form of constant sectional curvature c.

For each real namber c and each integer n > 1 there is (up to isometry) exactly one n-dimensional space form of constant sectional curvature c.

(i)

if c = 0, then ˜ M ⁿ⁺¹ (c) is Euclidean space E ⁿ⁺¹ .

(ii)

if c < 0, then ˜ M ⁿ⁺¹ (c) is Real hyperbolic space H ⁿ⁺¹ (c).

(iii)

if c > 0, then ˜ M ⁿ⁺¹ (c) is the sphere S ⁿ⁺¹ (c) in Euclidean space.

Let M ⁿ be a hypersurface in ˜ M ⁿ⁺¹ (c). A Riemannian manifold M ⁿ together with an isometric immersion f of M ⁿ into ˜ M ⁿ⁺¹ (c) is called a hypersurface of ˜ M ⁿ⁺¹ (c). Then f ∗ is one to one and g is given by

g = f ^∗ ˜ g,

where ˜ g is the Riemannian metric of ˜ M ⁿ⁺¹ (c). The metric g is called the induced metric on M ⁿ . Let ξ be a unit normal vector. Let ˜ ∇ and ∇ be the covariant differentiations of ˜ M ⁿ⁺¹ (c) from Riemannian metric ˜ g and from Riemannian metric g, respectively. Then we have the following

∇ ˜ f

X f ∗ Y = f ∗ ∇ X Y + g(AX, Y )ξ,

· · · Gauss formula

∇ ˜ f

X ξ = −f ∗ AX,

· · · Weingarten formula where

˜

g(ξ, ξ) = 1, ˜ g(f ∗ X, ξ) = 0.

In the following, let A denote the second fundamental form. Then we have R(X, Y )Z = c{g(Y, Z)X − g(X, Z)Y } + g(AY, Z)AX − g(AX, Z)AY,

· · · Gauss equation

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(∇ X A)Y = (∇ Y A)X.

· · · Codazzi equation Now M ⁿ is called a locally symmetric space if the curvature tensor R of M ⁿ satisfies

∇R = 0.

For example, a piece of the product of two spaces of constant curvature is a locally symmetric space. Let S be the Ricci tensor of M ⁿ . Then it is naturally considered the question under the weaker condition of

∇S = 0

(from now on, we call M ⁿ a hypersurface with parallel Ricci tensor) than ∇R = 0. With regard to this, P.J.Ryan proved the following: If M ⁿ is not constant curvature c and if ∇S = 0 on M ⁿ , then M ⁿ is an open subset of one of the product of space forms ˜ M ^k (c 1 ) × M ˜ ^n−k (c 2 ), 1 ≤ k ≤ n, or c = 0 and rankA = 2 on M ⁿ , where c 1 and c 2 are constant curvatures.

The Ricci tensor S is called the cyclic Ricci parallel tensor if S satisfies S(∇ X S)(Y, Z ) = 0

for any X , Y and Z tangent to M ⁿ .

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

Theorem. Let M ⁿ be a hypersurface with cyclic Ricci parallel tensor in a real space form M ˜ ⁿ⁺¹ (c) of constant sectional curvature c. If S(∇ X S)(Y, Z) = 0 and trace A = constant , then

∇A = 0 and a M ⁿ is a piece of the two spaces of constant curvature which is a locally symmetric space.

Let ˜ M ⁿ⁺¹ (c) be an (n + 1)-dimensional space form, i.e., a Riemannian manifold of constant sectional curvature, say, c. Let f : M → M ˜ be an isometric immersion of an n - dimensional Riemannian manifold M into ˜ M (c). For simplicity, we say that M is a hypersurface immersed in ˜ M and, for all local formulas and computations, we may consider f as an imbedding and thus identify x ∈ M with f (x) ∈ M ˜ . The tangent space T x (M ) is identified with a subspace of the tangent space T x ( ˜ M ), and the normal space T _x ^⊥ is the subspace of T x ( ˜ M ) consisting of all X ∈ T x ( ˜ M ) which are orthogonal to T x (M ) with respect to the Riemannian metric g.

For an arbitrary point x 0 ∈ M , we may choose a field of unit normal vectors ξ defined in a neighborhood U of x 0 . The second fundamental form h and the corresponding symmetric opreator A are defined and related to covariant differentiations ˜ ∇ and ∇ in ˜ M and M , respectively, by the following formulas:

(1) ∇ ˜ X Y = ∇ X Y + h(X, Y ),

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(2) ∇ ˜ X ξ = −AX,

where X and Y are vecter fields tangent to M . We rewrite the Gauss equation as follows

(3) R(X, Y ) = cX ∧ Y + AX ∧ AY, X, Y ∈ T x (M ),

where X ∧ Y denotes the skew-symmetric endomorphism of T x (M ) defined by (X ∧ Y )Z = g(Y, Z )X − g(X, Z)Y .

Also,we have the Codazzi equaton as Introduction (4) (∇ X A)(Y ) = (∇ Y A)(X ).

For any point x of U the (0,2)-type Ricci tensor S of M is defined by

S(X, Y ) = X n i=1

g(R(X, e i )e i , Y ).

(5)

Then from (3) we have

S(X, Y ) = (n − 1)cg(X, Y ) + (traceA)g(AX, Y ) − g(A ² X, Y ).

(6)

By using Gauss equation and this definition, we also have the following (∇ X S)(Y, Z) = ∇ X (S(Y, Z)) − S(∇ X Y, Z) − S(Y, ∇ X Z) (7)

= ∇ X ((n − 1)cg(Y, Z ) + (traceA)g(AY, Z) − g(A ² Y, Z))

− ((n − 1)cg(∇ X Y, Z) + (traceA)g(A∇ X Y, Z) − g(A ² ∇ X Y, Z ))

− ((n − 1)cg(Y, ∇ X Z) + (traceA)g(AY, ∇ X Z) − g(A ² Y, ∇ X Z))

= (traceA)g((∇ X A)Y, Z) − g((∇ X A ² )Y, Z ).

From (6) we have

(∇ X S)(Y, Z) + (∇ Y S)(Z, X) + (∇ Z S)(X, Y ) (8)

= (traceA)g((∇ X A)Y, Z) − g((∇ X A ² )Y, Z ) + (traceA)g((∇ Y A)Z, X ) − g((∇ Y A ² )Z, X ) + (traceA)g((∇ Z A)X, Y ) − g((∇ Z A ² )X, Y )

= 3(traceA)g((∇ _X A)Y, Z) − g((∇ _X A ² )Y, Z)

− g((∇ Y A ² )Z, X ) − g((∇ Z A ² )X, Y ).

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By this equation and Codazzi equation we obtein

(9) A(∇ X A)Y = 3(traceA)

2 (∇ X A)Y − (∇ X A)AY − (∇ Y A)AX.

Proposition 1 (P.J Ryan) Let M be a hypersurface of dimension > 2 in a real space form of constant curvature c. If M is not of constant curvature c and if ∇S = 0 on M , then M is an open subset of one of the standard examples or c = 0 and rankA = 2 on M .

Proposition 2 (P.J Ryan) Suppose that traceA is constant and ∇S = 0 ( S is the Ricci tensor). Then ∇A = 0.

Proposition 3 T λ

is differentiable.

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

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[1]Y.Matsuyama,Minimal Submanifolds in S ^N and R ^N ,Math.Z.175(1980),275-282

[2]Patrick J.Ryan,Homogeneity and Some Curvature Conditions for Hypersurfaces,Tohoku Math.

Journ.21(1969),363-388

[3]Patrick J.Ryan,Hypersurfaces with Parallel Ricci Tensor,Osaka J.Math.8(1971),251-259 [4]Katsumi Nomizu,On Hypersurfaces Satisfying a Certain Condition on The Curvature Ten- sor,Tohoku Math.Journ.20(1968),46-59

[5]Josrph Erbacher,Isometric Immersions of Constant mean Curvature and Triviality of The Nomal Connection,Nagoya Math.J.Vol45(1971),139-165

Department of mathmatices Chuo University 1-13-27 Kasuga,Bunkyo-ku Tokyo 112-8551,Japan

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