Sectional curvatures of holomorphic planes on a real hypersurface inPn(C)
Sectional curvatures of holomorphic planes on a real hypersurface inPn(C)
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Pn(C) 中实超曲面上全纯平面的截面曲率
DOI：
10.1007/bf01450843
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发表时间：
1987
影响因子：
1.4
通讯作者：
M. Kimura
中科院分区：
文献类型：
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作者：
M. Kimura
在研究Kfihler流形的微分几何性质时，全纯截曲率是一个重要的不变量。特别是具有常全纯截面曲率的复空间形式，是K~thler流形中最基本的例子。最近，ROS通过全纯Pinching得到了具有平行第二基本形式的K/Ihler子流形的一个刻画[8]。在本文中，我们考虑了关于复射影空间中的实超曲面的一个类似不变量。设P“(~)是n维常全纯截面曲率为4的复射影空间，M是Pn(IE)中的实超曲面。我们用H表示M上全纯2-平面的截面曲率。首先，我们研究了H为常数的实超曲面。设N是点xem上的单位法矢，J是Pn(IE)上的标准复结构。则~=-JN是XeM上的切向量.我们说，如果存在由复超平面W-i(IE)构成的M的叶状结构，则M是定则的.实空间形式的直纹子流形在[I]中被定义。我们有
The holomorphic sectional curvature is an important invariant when investigating the differential geometric properties of Kfihler manifolds. Especially, complex space forms, which have constant holomorphic sectional curvature, are the most fundamental examples among the K~ thler manifolds. Recently, Ros got a characterization of K/ihler submanifolds with parallel second fundamental form by holomorphic pinching [8].In this paper, we consider an analogous invariant with respect to the real hypersurfaces in a complex projective space. Let P"(~) be an n-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4, and let M be a real hypersurface in Pn (IE). We denote by H the sectional curvature of a holomorphic 2-plane on M. First, we study the real hypersurfaces on which H is constant. Let N be a unit normal vector at a point xeM, and let J be the canonical complex structure on Pn (IE). Then~=-JN is a tangent vector at xe M. We say that M is ruled if there is a foliation of M by complex hyperplanes W-I (IE). A ruled submanifold of a real space form is defined in [i]. We have