## 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 treatment introduces all the basics of the subject, and along with some classical results and hard-to-find proofs, presents new proofs of several recent results. Appendices furnish the necessary background material, exercises give readers practice in using the techniques, and discussion of open problems stimulates readers' interest in the field. |

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### Contents

Introduction | 1 |

Basics of submanifold theory in space forms | 7 |

21 The fundamental equations for submanifolds of space forms | 8 |

22 Models of space forms | 14 |

23 Principal curvatures | 17 |

24 Totally geodesic submanifolds of space forms | 20 |

25 Reduction of the codimension | 22 |

26 Totally umbilical submanifolds of space forms | 24 |

54 Homogeneous isoparametric submanifolds | 161 |

55 Isoparametric rank | 168 |

56 Exercises | 174 |

Rank rigidity of submanifolds and normal holonomy of orbits | 177 |

61 Submanifolds with curvature normals of constant length and rank of homogeneous submanifolds | 178 |

62 Normal holonomy of orbits | 191 |

63 Exercises | 198 |

Homogeneous structures on submanifolds | 201 |

27 Reducibility of submanifolds | 27 |

28 Exercises | 30 |

Submanifold geometry of orbits | 33 |

31 Isometric actions of Lie groups | 34 |

32 Polar actions and srepresentations | 41 |

33 Equivariant maps | 52 |

34 Homogeneous submanifolds of Euclidean space | 56 |

35 Homogeneous submanifolds of hyperbolic spaces | 58 |

36 Second fundamental form of orbits | 61 |

37 Symmetric submanifolds | 64 |

38 Isoparametric hypersurfaces in space forms | 81 |

39 Algebraically constant second fundamental form | 89 |

310 Exercises | 91 |

The Normal Holonomy Theorem | 95 |

41 Normal holonomy | 96 |

42 The Normal Holonomy Theorem | 106 |

43 Proof of the Normal Holonomy Theorem | 108 |

44 Some geometric applications of the Normal Holonomy Theorem | 116 |

45 Further remarks | 131 |

46 Exercises | 134 |

Isoparametric submanifolds and their focal manifolds | 139 |

51 Submersions and isoparametric maps | 140 |

52 Isoparametric submanifolds and Coxeter groups | 143 |

53 Geometric properties of submanifolds with constant principal curvatures | 157 |

71 Homogeneous structures and homogeneity | 202 |

72 Examples of homogeneous structures | 208 |

73 Isoparametric submanifolds of higher rank | 214 |

74 Exercises | 220 |

Submanifolds of Riemannian manifolds | 223 |

81 Submanifolds and the fundamental equations | 224 |

82 Focal points and Jacobi vector fields | 225 |

83 Totally geodesic submanifolds | 230 |

84 Totally umbilical submanifolds and extrinsic spheres | 236 |

85 Symmetric submanifolds | 240 |

86 Exercises | 241 |

Submanifolds of Symmetric Spaces | 243 |

92 Totally umbilical submanifolds and extrinsic spheres | 252 |

93 Symmetric submanifolds | 256 |

94 Submanifolds with parallel second fundamental form | 266 |

95 Homogeneous hypersurfaces | 269 |

96 Exercises | 280 |

Basic material | 281 |

A2 Lie groups and Lie algebras | 291 |

A3 Homogeneous spaces | 299 |

A4 Symmetric spaces and flag manifolds | 302 |

313 | |

331 | |

### Common terms and phrases

ambient space autoparallel Cartan classification codimension cohomogeneity compact constant curvature constant principal curvatures curvature normals decomposition defined denote dimension eigendistributions eigenvalues embedding Euclidean space Exercise exists extrinsic sphere focal manifold hence holonomy group holonomy tube homogeneous hypersurfaces homogeneous space homogeneous submanifold hyperbolic space hypersurfaces implies inner product invariant isometry isomorphic isoparametric hypersurfaces isoparametric submanifold isotropy representation Killing vector field Lemma Let G Lie algebra Lie group linear subspace mean curvature normal bundle normal holonomy group Normal Holonomy Theorem normal space normal vector field orthogonal parallel normal field parallel normal vector parallel second fundamental parallel transport polar principal curvatures principal orbit proof R-spaces rank respect Riemannian manifold Riemannian metric Riemannian symmetric space s-representation second fundamental form Section semisimple shape operator simply connected slice representation SO(n subalgebra submanifold of Euclidean submanifolds with constant symmetric submanifold tangent space totally geodesic totally geodesic submanifold V-parallel vector space

### References to this book

Differential Geometry and Its Applications Oldrich Kowalski,Demeter Krupka,Olga Krupkova No preview available - 2008 |