ON REAL HYPERSURFACES OF A COMPLEX SPACE FORM WITH ^-PARALLEL RICCI TENSOR
ON REAL HYPERSURFACES OF A COMPLEX SPACE FORM WITH ^-PARALLEL RICCI TENSOR
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具有^-平行RICCI张量的复杂空间形式的实超曲面
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发表时间：
1990
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通讯作者：
Young JinSuh
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作者：
Young JinSuh
设M_n(C)表示具有常全纯截面曲率c的n维复空间型。众所周知，一个完备的、单连通的复空间型由复射影空间Cpn、复欧氏空间Cn或复双曲空间Chn组成，满足c>；0，c=0或c<；0。本文考虑CPn或Chn的实超曲面M。对CPN的实超曲面的研究是由Takagi[10]开创的，他证明了CPN的所有齐次超曲面可以分为Au A2、B、C、D和E型六种类型，并且证明了如果Cpn的实超曲面M有两个或三个不同的常主曲率，则M局部同余于Au A2和B型齐次超曲面之一([11])。最近，为了给出CPN中AltA2和B型齐次超曲面的另一个刻画，Kimura和Maeda[6]引入了^-平行第二基本形式的概念，它由g((FXA)Y，Z)=0定义，对任何垂直于结构向量场GB的向量场X，Y和Z，其中A是M在CPN中的第二基本形式，g和V分别表示诱导黎曼度量和诱导黎曼联络。另一方面，许多作者(Berndt[1]，Montiel[8]，Montiel和Romero[9])也研究了Chn的实超曲面。Berndt[1]利用有关焦点集的一些结果证明了以下结论。
Let Mn(c) denote an n-dimensional complex space form with constant holomorphic sectional curvature c. It is well known that a complete and simplyconnected complex space form consists of a complex projective space CPn, a complex Euclidean space Cn or a complex hyperbolic space CHn, according as c>0, c=0 or c<0. In this paper we consider a real hypersurface M of CPn or CHn. The study of real hypersurfaces of CPn was initiated by Takagi [10], who proved that all homogeneous hypersurfaces of CPn could be divided into six types which are said to be of type Au A2, B, C, D and E. Moreover, he showed that if a real hypersurface M of CPn has two or three distinctconstant principal curvatures, then M is locally congruent to one of the homogeneous ones of type Au A2 and B ([11]). Recently, to give another charac terization of homogeneous hypersurfaces of type Alt A2 and B in CPn Kimura and Maeda [6] introduced the notion of an ^-parallel second fundamental form, which was defined by g((FxA)Y, Z)=0 for any vector fields X, Y and Z orthogonal to the structure vector field£,where A means the second fundamental form of M in CPn, and g and V denote the induced Riemannian metric and the induced Riemannian connection, respectively. On the other hand, real hypersurfaces of CHn have also been investigated by many authors (Berndt [1], Montiel [8], Montiel and Romero [9]). Using some results about focal sets, Berndt [1] proved the following.