周期的リッチ準対称テンソルを持つケーラー超曲面 Kaehler hypersurfaces with cyclic Ricci
semi-symmetric tensor
数学専攻 横山 教史 YOKOYAMA Norifumi Let M be a Kaehler hypersurface in a complex space form M˜n+1(c) of the complex dimension n+ 1 with constant holomor- phic sectional curvaturec. Ifc > 0 , then ˜Mn+1(c) is the complex projective spacePn+1(c) with the Fubini-Study metric of constant holomorphic sectional curvature c. If c <0 , then ˜Mn+1(c) is the open unit ball Dn+1(c) in the complex Euclidean space Cn+1 with the Bergman metric of constant holomorphic sectional curvature c. If c = 0 , then ˜Mn+1(c) is Cn+1.
Let S and R be the Ricci tensor and the curvature tensor of M , respectively . P. J. Ryan [1] proved that if M has the Ricci semi-symmetric tensor, i.e. (R(X, Y)S)Z = 0 , then
(i) when c = 0 , A2 is a multiple of I;
(ii) when c = 0 , the non-zero eigenvalues of A2 are equal .
Moreover , the Ricci tensor S is called theparallel Ricci tensor if M satisfies the condition of ∇S = 0 . He also proved that∇S = 0 and M is Einstein are equivalent .
On the other hand , Nomizu and Smyth [3] classified the com- plex hypersurfaces with parallel Ricci tensor . They proved that if ∇S = 0 on M , then
(i) when c = 0 , M is totally geodesic ;
(ii) when c = 0 , either M is totally geodesic or M is locally the complex quadric in ˜Mn+1(c) , the latter case arising only when c >0 .
The Ricci tensor S is called the cyclic Ricci semi-symmetric
tensor if M satisfies the condition of S(R(X, Y)S)Z = 0 , i.e. , (R(X, Y)S)Z + (R(Y, Z)S)X + (R(Z, X)S)Y = 0 for any X ,Y and Z tangent to M .
The purpose of this paper is to classify Kaehler hypersurfaces with cyclic Ricci semi-symmetric tensor.
Theorem. Let M be a connected Kaehler hypersurface with cyclic Ricci semi-symmetric tensor in a complex space form M˜n+1(c) of constant holomorphic sectional curvature c . If c = 0, then M is Einstein .
The following Proposition will be used frequently in my paper . Proposition 1 IfR˜ and R denote the Riemannian curvature ten- sors of M˜ and M , respectively , then for any vector fields X ,Y ,Z and W on U(x)
(i) the ξ-component of R(X, Y˜ )W is given by
g((∇XA)Y −(∇YA)X−s(X)J AY +s(Y)J AX, W)ξ; the expression for the Jξ-component of R(X, Y˜ )W is obtained on replacing A by JA and ξ by Jξ.
(ii) 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(J AX, Z)g(J AY, W)
−g(J AX, W)g(J AY, Z)} (Gauss equation) .
Proposition 2 If M˜ is of constant holomorphic sectional curva- ture c , then for any pair of vectors X and Y tangent to M at a point of U(x) , we have the equations
(i) (∇XA)Y −(∇YA)X −s(X)J AY + s(Y)J AX = 0 (Codazzi equation) , (ii) S(X, Y) =−2g(A2X, Y) + (n+1)c2 g(X, Y),
where S is the Ricci tensor of M .
Furthermore, we introduce some expression to use in this paper . Let A and J be the second f undamental f orm and Kaehler structure on ˜Mn+1(c) . Then for any vector fields X , Y and Z tangent to M on U(x) , we have
(1) AJ = −J A , (2) R(X, Y˜ )W = c
4{g(Y, W)X −g(X, W)Y +g(J Y, W)J X −g(J X, W)J Y +2g(X, J Y)J W},
(3) R(X, Y)W = ˜R(X, Y)W
+{g(AY, W)AX −g(AX, W)AY}
+{g(J AY, W)J AX−g(J AX, W)J AY}, (4) R(X, Y)W = c
4{g(Y, W)X −g(X, W)Y +g(J Y, W)J X −g(J X, W)J Y +2g(X, J Y)J W}
+g(AY, W)AX −g(AX, W)AY
+g(J AY, W)J AX−g(J AX, W)J AY ,
——— Gauss equation
(5) R(X, Y)Z + R(Y, Z)X +R(Z, X)Y = 0,
——— f irst Bianchi identity
(6) SX = n+ 1
2 cX −2A2X .
The author would like to express the will of great thanks for indications and cooperations of Professor Y. Matsuyama on writ- ing this paper .
参考文献
[1] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol II, Inter science Tracts No.15, John Wiley and Sons, New York(1963).
[2] Y. Matsuyama.,Minimal Submanifolds inSN and RN, Math.
Z. 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), 498-521.
[4] P. J. Ryan., A class of complex hypersurfaces, Colloquium Mathmaticum 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
