Metric Foliations and Curvature (eBook)

Detlef Gromoll, Gerard Walschap (Autoren)

eBook Download: PDF

2009 | 2009
VIII, 176 Seiten
Springer Basel (Verlag)
978-3-7643-8715-0 (ISBN)

Infos

Riemannian manifolds, particularly those with positive or nonnegative curvature, are constructed from only a handful by means of metric fibrations or deformations thereof. This text documents some of these constructions, many of which have only appeared in journal form. The emphasis is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.

Inhalt

Contents 6
Preface 8
Submersions, Foliations, and Metrics 10
1.1 Notation and basic geometric concepts 10
1.2 Metric foliations and Riemannian submersions 14
1.3 Horizontal lifts and transversal holonomy 19
1.4 The fundamental tensors of a submersion 23
1.5 Curvature relations 33
1.6 Projectable Jacobi fields 41
1.7 The Riccati equation for Jacobi fields 44
1.8 The dual foliation 49
1.9 Basic identities 53
Basic Constructions and Examples 54
2.1 General vertical warping 54
2.2 Warped products 65
2.3 Homogeneous submersions 72
2.4 Left-invariant metrics on Lie groups 81
2.5 The Aloff-Wallach examples 91
2.6 Bi-quotients of Lie groups 96
2.7 Associated bundles 101
2.8 Fat bundles 112
Open Manifolds of Nonnegative Curvature 118
3.1 Convex sets in Riemannian manifolds 118
3.2 The soul construction 122
3.3 The topological structure of 129
3.4 The Sharafutdinov retraction 131
3.5 The metric projection onto the soul 135
3.6 The metric structure of bundles with 139
Metric Foliations in Space Forms 144
4.1 Isoparametric foliations 144
4.2 Metric fibrations of Euclidean space 149
4.3 Metric foliations of spheres 158
4.4 Geometry of the tangent bundle 166
4.5 Compact space forms of nonpositive curvature 170
Bibliography 174
Index 182

Produkt-Details

• Erscheint lt. Verlag: 28.3.2009
• Reihe/Serie: Progress in Mathematics
• Reihe/Serie: Progress in Mathematics
• Zusatzinfo: VIII, 176 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Curvature • Differential Geometry • foliation • manifold • Riemannian manifold • space form
• ISBN-10: 3-7643-8715-7 / 3764387157
• ISBN-13: 978-3-7643-8715-0 / 9783764387150