Submanifolds and Holonomy
Submanifolds and Holonomy
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DOI：
10.1201/9780203499153
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发表时间：
2016-02
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影响因子：
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通讯作者：
J. Berndt;S. Console;C. Olmos
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文献类型：
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作者：
J. Berndt;S. Console;C. Olmos
空间中子流形理论的基础形式空间形式子流形的基本方程空间形式的模型主曲率空间形式的全测地子流形余维全脐子流形的约化空间形式的全脐子流形归约轨道的几何李群的等距作用等距作用的存在极作用和S表示等变映射欧几里德空间的齐次子流形双曲空间的齐次子流形轨道对称子流形的第二基本形式轨道对称子流形的基本形式空间中的等参超曲面第二基本形式正规完整定理的正规完整定理的几何证明常主曲率齐次等参子流形的几何性质等参子流形等参子流形的几何性质等参子流形的等参秩刚性和等长等秩轨道子流形的法向完整齐次子流形上轨道齐次结构的正规完整齐次结构实例复子流形的高阶等参子流形算子在平行流形中具有某些常特征值的叶的类极性质全孔管的正则叶变换在具有非传递法向完整的Cn的复子流形上的应用完整理论Berger-Simons完整理论完整系统Simons完整定理Berger完整定理斜挠子流形定理等距不动点集和齐次子流形自然约化空间完全斜一型值在李代数中的值导出的2-型值在李代数中导出的2-型值在自然约化空间黎曼流形的子流形和基本方程完全测地子流形的焦点和Jacobi场完全测地子流形和外球对称子流形全脐子流形和外部子流形具有平行第二基本形式作用的对称子流形型极作用-秩一极作用-高阶超极作用-高阶同调一作用-非紧致型抛物子代数对称空间上具有常主曲率的高阶超曲面极作用非紧型抛物子代数的极作用无奇异轨道的极作用双曲空间上无奇异轨道的极作用同调一作用-具有常主曲率的高阶超曲面附录：基本材料练习出现在每一章的末尾。
Basics of Submanifold Theory in Space Forms The fundamental equations for submanifolds of space forms Models of space forms Principal curvatures Totally geodesic submanifolds of space forms Reduction of the codimension Totally umbilical submanifolds of space forms Reducibility of submanifolds Submanifold Geometry of Orbits Isometric actions of Lie groups Existence of slices and principal orbits for isometric actions Polar actions and s-representations Equivariant maps Homogeneous submanifolds of Euclidean spaces Homogeneous submanifolds of hyperbolic spaces Second fundamental form of orbits Symmetric submanifolds Isoparametric hypersurfaces in space forms Algebraically constant second fundamental form The Normal Holonomy Theorem Normal holonomy The normal holonomy theorem Proof of the normal holonomy theorem Some geometric applications of the normal holonomy theorem Further remarks Isoparametric Submanifolds and Their Focal Manifolds Submersions and isoparametric maps Isoparametric submanifolds and Coxeter groups Geometric properties of submanifolds with constant principal curvatures Homogeneous isoparametric submanifolds Isoparametric rank Rank Rigidity of Submanifolds and Normal Holonomy of Orbits Submanifolds with curvature normals of constant length and rank of homogeneous submanifolds Normal holonomy of orbits Homogeneous Structures on Submanifolds Homogeneous structures and homogeneity Examples of homogeneous structures Isoparametric submanifolds of higher rank Normal Holonomy of Complex Submanifolds Polar-like properties of the foliation by holonomy tubes Shape operators with some constant eigenvalues in parallel manifolds The canonical foliation of a full holonomy tube Applications to complex submanifolds of Cn with nontransitive normal holonomy Applications to complex submanifolds of CPn with nontransitive normal holonomy The Berger-Simons Holonomy Theorem Holonomy systems The Simons holonomy theorem The Berger holonomy theorem The Skew-Torsion Holonomy Theorem Fixed point sets of isometries and homogeneous submanifolds Naturally reductive spaces Totally skew one-forms with values in a Lie algebra The derived 2-form with values in a Lie algebra The skew-torsion holonomy theorem Applications to naturally reductive spaces Submanifolds of Riemannian Manifolds Submanifolds and the fundamental equations Focal points and Jacobi fields Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds of Symmetric Spaces Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds with parallel second fundamental form Polar Actions on Symmetric Spaces of Compact Type Polar actions - rank one Polar actions - higher rank Hyperpolar actions - higher rank Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Polar Actions on Symmetric Spaces of Noncompact Type Dynkin diagrams of symmetric spaces of noncompact type Parabolic subalgebras Polar actions without singular orbits Hyperpolar actions without singular orbits Polar actions on hyperbolic spaces Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Appendix: Basic Material Exercises appear at the end of each chapter.