Submanifolds and Holonomy

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CRC Press, Apr 28, 2003 - Mathematics - 352 pages
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With 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
References
313
Index
331
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mathnetbase: Mathematics Online
Submanifolds and Holonomy. Jurgen Berndt Sergio Console Carlos Olmos. Read it Online! With special emphasis on new techniques based on the holonomy of the ...
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Submanifolds and Holonomy
Submanifolds and Holonomy. Authors: Jurgen Berndt; Sergio Console; Carlos Olmos. ISBN: 978-0-203-49915-3 (electronic). No. of pages: 336 ...
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洋図書2003/07
159, Submanifolds and holonomy / J鰕rgen Berndt, Sergio Console, and Carlos Olmos. -- Chapman & Hall/CRC, c2003. -- (Research notes in mathematics ; 434). ...
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