擬リッチ平行テンソルを持つ超曲面 Hypersurfaces
with nearly Ricci parallel tensor
数学専攻 小野 峻 Takashi ONO
Let ˜Mn+1(c) be an (n+ 1)-dimensional real space form of constant sectional curvaturec(i.e. complete, simply connected Riemannian manifold of constant sectional curvature, say, c). For each real namber c and each integer n > 1 there is (up to isometry) exactly onen-dimensional real space form of constant sectional curvature c.
The real space forms are
(i) if c= 0, then ˜Mn+1(c) is Euclidean space En+1.
(ii) if c <0, then ˜Mn+1(c) is Real hyperbolic space Hn+1(c).
(iii) if c >0, then ˜Mn+1(c) is the sphere Sn+1(c) in Euclidean space.
Let f be isometric immersion of Mn as a hypersurface in ˜Mn+1(c). Let R and ∇be the curvature tensor of Mn and the covariant differentiation in Mn, respectively. Mn is called a locally symmetric space if the curvature tensor R of Mn satisfies
∇R = 0.
For example, a piece of the product of two spaces of constant curvature is a locally symmetric space. LetS be the Ricci tensor ofMn. Then it is naturally considered the question under the weaker condition of
∇S = 0
(from now on, we call Mn a hypersurface with parallel Ricci tensor) than
∇R = 0. With regard to this, P.J.Ryan proved the following: If Mn is not constant curvature cand if ∇S = 0 on Mn, thenMn is an open subset of one of the product of space forms ˜Mk(c1) × M˜n−k(c2), 1 ≤ k ≤ n, or c = 0 and rankA = 2 onMn, wherec1 and c2 are constant curvatures.
The Ricci tensor S is called the nearly Ricci parallel tensor if S satisfies (∇XS)X = 0
for any X tangent to Mn.
The purpose of this paper is to classify hypersurfaces with nearly 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 Mn be a hypersurface with nealy Ricci parallel tensor in a real space form M˜n+1(c) of constant sectional curvature c. If (∇XS)X = 0 and traceA = 0 , then ∇A = 0 and Mn is a piece of two spaces of constant curvature which is a locally symmetric space.
We introduce some expressions to use in this paper.
Let ˜Mn+1(c) be an (n+ 1)-dimensional space form, i.e., a Riemannian man- ifold of constant sectional curvature, say, c. Let φ :M →M˜ be an isometric immersion of an n-dimensional Riemannian manifold M into ˜M. For sim- plicity, we say that M is a hypersurface immersed in ˜M and, for all local formulas and computations, we may consider φ as an imbedding and thus identify x∈M with φ(x)∈M˜. The tangent space Tx(M) is identified with a subspace of the tangent spaceTx( ˜M), and the normal spaceTx⊥is the subspace of Tx( ˜M) consisting of all X ∈ Tx( ˜M) which are orthogonal to Tx(M) with respect to the Riemannian metric g.
For an arbitrary pointx0 ∈M, we may choose a field of unit normal vectors ξ defined in a neighborhood U of x0. 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) ∇˜XY =∇XY +h(X, Y),
(2) ∇˜Xξ =−AX,
where X and Y are vecter fields tangent to M. The Gauss equation is:
(3) R(X, Y) = cX∧Y +AX∧AY, X, Y ∈Tx(M),
whereX∧Y denotes the skew-symmetric endomorphism of Tx(M) defined by (X∧Y)Z =g(Y, Z)X−g(X, Z)Y.
The Codazzi equation is expressed by
(4) (∇XA)(Y) = (∇YA)(X).
Moreover, we denote the (0,2)-type Ricci tensor of M byS. For any point x of U, S is defined by
(5) S(X, Y) = Xn
i=1
g(R(ei, X)Y, ei),
where {e1, . . . , ei} is an orthonormal basis of the tangent spaceTx(M). Using Gauss equation (3) and assuming thatS(X, Y) = g(SX, Y), we obtain
(6) SX = (n−1)cX + (traceA)AX−A2X,
for any X tangent to M onU. We put
Tλi(x0) ={X ∈Tx0Mn |AX =λiX}.
Furthemore, the following Propositions will be used frequently in my paper.
Proposition 1 (P.J Ryan[3]) 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 rank A= 2 on M.
Proposition 2 (P.J Ryan[3]) Suppose trace A is constant and ∇S = 0 (S is the Ricci tensor). Then ∇A= 0.
Proposition 3 Tλi is 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.
参考文献
[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 SN and RN, Mathematische Zeitschrift 175 (1980), 275-282.
[3] P. J. Ryan,Hypersurface with parallel Ricci tensor, Osaka Math. J.8(1971), 251-259
[4] P. J. Ryan,Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math.J.21(1969), 363-388.
[5] K. Nomizu,On hypersurfaces satisfing a certain condition on the curvature tensor,Tohoku Math.J.20(1968), 46-59.
[6] K. Nomizu and B. Snyth, A formula of Simons’ type and hypersurfaces with constant mean curvature, J. Differential geometry.3(1969), 367-377 [7] T. Yamada, On hypersurfaces of a real space form, thesis(2001).
[8] K. Toda, Kaehler hypersurfaces with nearly Ricci parallel tensor, the- sis(2006).
Department of Mathematics, Chuo University
1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan
