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1 Introduction 1
2 Basics of submanifold theory in space forms 7
2.1 The fundamental equations for submanifolds of space forms .... 8
2.2 Models of space forms ........ ................ 14
2.3 Principal curvatures........................ 17
2.4 Totally geodesic submanifolds of space forms ........... 20
2.5 Reduction of the codimension.................... 22
2.6 Totally umbilical submanifolds of space forms ..... ..... 24
2.7 Reducibility of submanifolds.............. .... 27
2.8 Exercises .......... ..................... 30
3 Submanifold geometry of orbits 33
3.1 Isometric actions of Lie groups................. 34
3.2 Polar actions and s-representations................. 41
3.3 Equivariant maps .......................... 52
3.4 Homogeneous submanifolds of Euclidean space . ......... 56
3.5 Homogeneous submanifolds of hyperbolic spaces ........ . 58
3.6 Second fundamental form of orbits................. 61
3.7 Symmetric submanifolds ....................... 64
3.8 Isoparametric hypersurfaces in space forms ......... ... . 81
3.9 Algebraically constant second fundamental form .......... 89
3.10 Exercises ............................. . 91
4 The Normal Holonomy Theorem 95
4.1 Normal holonomy .......................... 96
4.2 The Normal Holonomy Theorem ............ .. 106
4.3 Proof of the Normal Holonomy Theorem . ............ 108
4.4 Some geometric applications of the Normal Holonomy Theorem . 116
4.5 Further remarks............................ 131
4.6 Exercises ..... ............... ......... .. 134
5 Isoparametric submanifolds and their focal manifolds 139
5.1 Submersions and isoparametric maps ................140
5.2 Isoparametric submanifolds and Coxeter groups ......... 143
5.3 Geometric properties of submanifolds with constant principal curva-
tures. .. . .. . .. . .. . .. . . .. .. . . .. . .. . .. . . . 157
5.4 Homogeneous isoparametric submanifolds ................ 161
5.5 Isoparametric rank.......................... 168
5.6 Exercises ............................... 174
6 Rank rigidity of submanifolds and normal holonomy of orbits 177
6.1 Submanifolds with curvature normals of constant length and rank of
homogeneous submanifolds ..................... 178
6.2 Normal holonomy of orbits ..................... 191
6.3 Exercises ............................... 198
7 Homogeneous structures on submanifolds 201
7.1 Homogeneous structures and homogeneity ............. 202
7.2 Examples of homogeneous structures ................ 208
7.3 Isoparametric submanifolds of higher rank ............. 214
7.4 Exercises ............................... 219
8 Submanifolds of Riemannian manifolds 223
8.1 Submanifolds and the fundamental equations ............ 224
8.2 Focal points and Jacobi vector fields ...... ........... 225
8.3 Totally geodesic submanifolds................... . 230
8.4 Totally umbilical submanifolds and extrinsic spheres ........ 236
8.5 Symmetric submanifolds ..................... ......... 240
8.6 Exercises ............................... 241
9 Submanifolds of Symmetric Spaces 243
9.1 Totally geodesic submanifolds.................... 243
9.2 Totally umbilical submanifolds and extrinsic spheres ........ 252
9.3 Symmetric submanifolds ....................... 256
9.4 Submanifolds with parallel second fundamental form ........ 266
9.5 Homogeneous hypersurfaces ................. ....... 269
9.6 Exercises ............................... 280
Appendix Basic material 281
A.1 Riemannian manifolds ........................ 281
A.2 Lie groups and Lie algebras........... ......... . 291
A.3 Homogeneous spaces ................... .......... 299
A.4 Symmetric spaces and flag manifolds ................ 302
References ............................................................. 313
Index .................................................................. 331
Library of Congress Subject Headings for this publication: Submanifolds, Holonomy groups