Riemannian Submersions and Related Topics

Maria Falcitelli, Anna Maria Pastore, Stere Ianus9812388966, 9789812388964, 9789812562333

For graduate students and researchers interested in differential geometry and theoretical physics, Falcitelli, Anna Maria Pastore (both U. of Bari, Italy) and Steve Ianus (U. of Bucharest, Romania) provide a systematic exposition of the basic tools on the theory of Riemannian submersions and its developments. Submersions are special tools in differential geometry, but also play a role in Riemannian geometry, especially when the manifolds involved carry an additional structure. They are in a sense a counterpart of immersions, but the corresponding theories and histories are quite different.

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Table of contents :
Riemannian Submersions and Related Topics……Page 4
Preface……Page 8
Contents……Page 12
1. Riemannian Submersions……Page 16
1.1 Riemannian Submersions……Page 17
1.2 The Hopf Fibration……Page 19
1.3 Fundamental Tensors and Fundamental Equations……Page 23
1.4 Other Examples……Page 32
1.5 Geodesics and O’Neill Theorem……Page 40
1.6 Clairaut Submersions……Page 43
2. Submersions with Totally Geodesic Fibres……Page 48
2.1 Riemannian Submersions with Complete Total Space……Page 49
2.2 Submersions with Totally Geodesic Fibres……Page 53
2.3 Riemannian Submersions from the Spheres……Page 59
2.4 The Uniqueness Theorem……Page 64
2.5 Submersions from the Complex Projective Spaces……Page 69
3. Almost Hermitian Submersions……Page 74
3.1 Almost Hermitian Manifolds……Page 75
3.2 Almost Hermitian Submersions……Page 79
3.3 Holomorphic Distributions in Kahler Submersions……Page 86
3.4 Curvature Properties……Page 90
3.5 Locally Conformal Kahler Manifolds……Page 96
3.6 Locally Conformal Kahler Submersions……Page 103
3.7 Submersions from Generalized Hopf Manifolds……Page 106
3.8 Almost Complex Conformal Submersions……Page 112
4. Riemannian Submersions and Contact Metric Manifolds……Page 118
4.1 Remarkable Classes of Contact Metric Manifolds……Page 119
4.2 Contact Riemannian Submersions……Page 124
4.3 Contact-complex Riemannian Submersions……Page 129
4.4 Complex-contact Riemannian Submersions……Page 131
4.5 Curvature Properties……Page 135
4.6 Riemannian Submersions Involving Quaternionic Kahler and 3-Sasakian Manifolds……Page 137
4.7 Regular f-structures and Riemannian Submersions……Page 144
5. Einstein Spaces and Riemannian Submersions……Page 158
5.1 Einstein Metrics on the Total Space of a Riemannian Submersion……Page 159
5.2 The Canonical Variation of the Metric in the Total Space……Page 163
5.3 Homogeneous Einstein Spaces……Page 168
5.4 Einstein Metrics on Principal Bundles……Page 177
5.5 Einstein Weyl Structures on Principal Bundles……Page 190
5.6 Einstein Weyl Structures on Hermitian and Sasakian Manifolds……Page 197
6. Riemannian Submersions and Submanifolds……Page 210
6.1 Submersions of CR-submanifolds……Page 211
6.2 Links Between Submanifolds of Sasakian and Kahler Manifolds……Page 216
6.3 Riemannian Submersions and Isometric Reflections with Respect to the Fibres……Page 224
7. Semi-Riemannian Submersions……Page 230
7.1 Semi-Riemannian Manifolds……Page 231
7.2 Semi-Riemannian Submersions. Examples…….Page 233
7.3 Lorentzian Submersions……Page 236
7.4 Submersions from Pseudo-hyperbolic Spaces……Page 238
7.5 Submersions with Totally Umbilical Fibres……Page 247
7.6 Semi-Riemannian Submersions with Minimal Fibres……Page 251
8.1 Gauge Fields, Instantons and Riemannian Submersions……Page 254
8.2 Einstein Equations and Kaluza–Klein Ansatz……Page 258
8.3 Generalized Nonlinear Sigma Model in Curved Space……Page 261
8.4 Horizontally Conformal Submersions and Gravity……Page 265
8.5 The Dirac Monopole and the Hopf Map……Page 266
8.6 Kaluza–Klein Monopole and Taub–NUT Instanton……Page 270
Bibliography……Page 272
Index……Page 290