Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics
Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics
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通过观察测地线的外在形状来表征复杂射影空间中的测地线超球面
DOI：
10.1007/pl00004625
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发表时间：
1997
影响因子：
0.8
通讯作者：
K. Ogiue
中科院分区：
文献类型：
--
作者：
S. Maeda;K. Ogiue
在某些情况下，可以通过观察子流形测地线的外在形状来了解子流形的形状 ([3])。例如：如果通过每个点的三个测地线在 E3 中是圆，则 E3 中的表面局部是球体。事实上，很容易看出，这样的测地线的切向量是形状算子的特征向量。本文的目的是研究Pn(C)的实超曲面的此类问题。令 Pn (C) 为具有恒定全纯截面曲率 4 的 Fubini-Study 度量的 n 维复射影空间，并令 M 为 Pn (C) 的实超曲面。那么 M 具有从 Pn (C) 的 Kaehler 结构导出的几乎接触度量结构 (;; Á; g)。 Pn (C) 中实超曲面的典型示例是齐次实超曲面，即，在射影酉群 PU (n+1) 的子群下给出的实超曲面。 R. Takagi ([9]) 对 Pn (C) 的齐次实超曲面进行分类，并证明 Pn (C) 的齐次实超曲面与 A1 类型的六个模型空间之一局部全等； A2； B; C; D 和 E。由于他的分类，Pn (C) 的所有齐次实超曲面都被实现为 1 级或 2 阶紧凑埃尔米特对称空间上的恒定半径的管（详细信息，请参见命题 A）。此外，我们发现齐次实超曲面的不同主曲率的数量为 2、3 或 5。
It is possible in some cases to know the shape of a submanifold by observing the extrinsic shape of geodesics of the submanifold ([3]). For example: A surface in E3 is locally a sphere if three geodesics through each point are circles in E3. In fact, it is easily seen that the tangent vector of such a geodesic is an eigenvector of the shape operator.The purpose of this paper is to study such a problem for real hypersurfaces of Pn (C). Let Pn (C) be an n-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4, and let M be a real hypersurface of Pn (C). Then M has an almost contact metric structure (;; Á; g) induced from the Kaehler structure of Pn (C). Typical examples of real hypersurfaces in Pn (C) are homogeneous real hypersurfaces, that is, real hypersurfaces given as orbits under subgroups of the projective unitary group PU (n+ 1). R. Takagi ([9]) classifies homogeneous real hypersurfaces of Pn (C) and he shows that a homogeneous real hypersurface of Pn (C) is locally congruent to one of the six model spaces of type A1; A2; B; C; D and E. Due to his classification, all homogeneous real hypersurfaces of Pn (C) are realized as tubes of constant radius over compact Hermitian symmetric spaces of rank 1 or 2 (for detail, see Proposition A). Moreover we find that the number of distinct principal curvatures of a homogeneous real hypersurface is 2, 3 or 5.