Submanifolds and HolonomyWith special emphasis on new techniques based on the holonomy of the normal connection, this book provides a modern, self-contained introduction to submanifold geometry. It offers a thorough survey of these techniques and their applications and presents a framework for various recent results to date found only in scattered research papers. The trea |
Contents
Ch 1 Introduction | 1 |
Ch 2 Basics of submanifold theory in space forms | 7 |
Ch 3 Submanifold geometry of orbits | 33 |
Ch 4 The Normal Holonomy Theorem | 95 |
Ch 5 Isoparametric submanifolds and their focal manifolds | 139 |
Ch 6 Rank Rigidity of submanifolds and normal holonomy of orbits | 177 |
Ch 7 Homogeneous structures on submanifolds | 201 |
Ch 8 Submanifolds of Riemannian manifolds | 223 |
Ch 9 Submanifolds of Symmetric Space | 243 |
Appendix Basic Material | 281 |
| 313 | |
List of Figures | 327 |
List of Tables | 329 |
| 331 | |
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Common terms and phrases
action acts assume called codimension coincides compact complex connected consider constant contained corresponding curvature normals curvature tensor curve decomposition defined denote determined differentiable dimension direction distribution embedding equation Euclidean space example Exercise exists extrinsic fact geometry given hence homogeneous hypersurfaces identity immersion implies induced invariant irreducible isometry isoparametric submanifold isotropy Killing Lemma Lie algebra Lie group linear locally maximal mean metric normal bundle normal space normal vector Note orbit orthogonal parallel parallel normal particular principal orbit projective proof Proposition Prove rank reducible REMARK representation respect restricted result Riemannian curvature Riemannian manifold second fundamental form shape operator smooth space forms sphere standard structure subgroup subspace symmetric space symmetric submanifold tangent Theorem totally geodesic totally geodesic submanifold vector field Þeld