THE RICCI TENSOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
THE RICCI TENSOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
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复杂二平面格拉斯曼尼亚实超曲面的 RICCI 张量
DOI：
10.4134/jkms.2007.44.1.211
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发表时间：
2007
影响因子：
4.5
通讯作者：
Suh Young
中科院分区：
文献类型：
--
作者：
Perez Juan De Dios;Suh Young
抽象的。本文首先从高斯方程引入复二平面格拉斯曼函数G 2 (C m +2 )中实超曲面M的曲率张量的完整表达式，并推导了G 2 (C m +2 )中M的Ricci张量的新公式。接下来我们证明在具有平行且可交换的Ricci 张量的复二平面格拉斯曼G 2 (C m +2 ) 中不存在任何Hopf 实超曲面。最后证明G 2 (C m +2 ) 中不存在爱因斯坦Hopf 超曲面。引言在复空间形式或四元空间形式的实超曲面的几何中，利用Co-dazzi方程可以很容易地证明不存在具有平行形状算子A的实超曲面。但是，如果我们考虑在这种空间形式中具有平行Ricci张量S的实超曲面，那么证明它不存在就不那么容易了。在 Hopf 超曲面类中，Kimura [7] 断言在复射影空间 C 中不存在任何真实的超曲面
Abstract. In this paper, flrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C