Riemannian geometry

T. J. Willmore9780198514923, 0198514921

Recent developments in the field of differential geometry have been so extensive that a new book with particular emphasis on current work in Riemannian geometry is clearly necessary. This new text brilliantly serves that purpose and includes an elementary account of twistor spaces that will interest both applied mathematicians and physicists. It presents recent developments in the theory of harmonic spaces, commutative spaces, and mean-value theorems previously only available in the source literature. The final chapter provides the only account available in book form of manifolds known as `Willmore surfaces’, illustrated by a series of computer-generated pictures. This book is sure to be welcomed by researchers, teachers, and students interested in the latest developments in differential geometry.

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Table of contents :
Cover……Page 1
Title page……Page 3
Date-line……Page 4
Dedication……Page 5
Preface……Page 7
Contents……Page 9
1.1 Preliminaries……Page 13
1.2 Differentiable manifolds……Page 14
1.3 Tangent vectors……Page 17
1.4 The differential of a map……Page 22
1.6 Submanifolds; the inverse and implicit function theorems……Page 26
1.7 Vector fields……Page 28
1.8 Distributions……Page 30
1.9 Exercises……Page 32
2.1 Tensor product……Page 33
2.2 Tensor fields and differential forms……Page 39
2.3 Exterior derivation……Page 40
2.4 Differential ideals and the theorem of Frobenius……Page 44
2.5 Orientation of manifolds……Page 46
2.6 Covariant differentiation……Page 50
2.7 Identities satisfied by the curvature and torsion tensors……Page 54
2.8 The Koszul connexion……Page 55
2.9 Connexions and moving frames……Page 60
2.10 Tensorial forms……Page 66
2.11 Exercises……Page 67
3.1 Riemannian metrics……Page 68
3.2 Identities satisfied by the curvature tensor of a Riemannian manifold……Page 70
3.3 Sectional curvature……Page 72
3.4 Geodesies……Page 77
3.5 Normal coordinates……Page 84
3.6 Volume form……Page 91
3.7 The Lie derivative……Page 94
3.8 The Hodge star operator……Page 96
3.9 Maxwell’s equations……Page 105
3.10 Consequences of the theorems of Stokes and of Hodge……Page 107
3.11 The space of curvature tensors……Page 111
3.12 Conformal metrics and conformal connexions……Page 117
3.13 Exercises……Page 125
4.1 Submanifolds……Page 127
4.2 The equations of Gauss, Codazzi, and Ricci……Page 133
4.3 The method of moving frames……Page 137
4.4 The theory of curves……Page 141
4.5 The theory of surfaces……Page 144
4.6 Surface theory in classical notation……Page 151
4.7 The Gauss-Bonnet theorem……Page 154
4.8 Exercises……Page 163
5.1 Complex structures on vector spaces……Page 164
5.2 Hermitian metrics……Page 167
5.3 Almost-complex manifolds……Page 169
5.4 Walker derivation……Page 176
5.5 Hermitian metrics and Kaehler metrics……Page 178
5.6 Complex manifolds……Page 180
5.7 Kaehler metrics in local coordinate systems……Page 182
5.8 Holomorphic sectional curvature……Page 186
5.9 Curvature and Chern forms of Kaehler manifolds……Page 189
5.10 Review of complex structures on vector spaces……Page 190
5.11 The twistor space of a Riemannian manifold……Page 192
5.12 The twistor space of $S^4$……Page 193
5.13 Twistor spaces of spheres……Page 196
5.14 Riemannian immersions and Codazzi’s equation……Page 197
5.15 Isotropic immersions……Page 199
5.16 The bundle of oriented orthonormal frames……Page 201
5.17 The relationship between the frame bundle and the twistor space……Page 202
5.18 Integrability of the almost-complex structures on the twistor space……Page 204
5.19 Interpretation of the integrability condition……Page 207
5.20 Exercises……Page 212
6.1 Harmonic and related metrics……Page 213
6.2 Normal coordinates……Page 214
6.3 The distance function $Omega$……Page 218
6.4 The discriminant function……Page 220
6.5 Geodesic polar coordinates……Page 222
6.6 Jacobi fields……Page 225
6.7 Definition of harmonic metrics……Page 235
6.8 Curvature conditions for harmonic manifolds……Page 240
6.9 Consequences of the curvature conditions……Page 246
6.10 Harmonic manifolds of dimension 4……Page 252
6.11 Mean-value properties……Page 258
6.12 Commutative metrics……Page 265
6.13 The curvature of Einstein symmetric spaces……Page 267
6.14 D’Atri spaces……Page 269
6.15 Exercises……Page 279
7.1 Introduction……Page 281
7.2 Surfaces in $E^3$……Page 282
7.3 Conformal invariance……Page 288
7.4 The Euler equation……Page 291
7.5 Willmore surfaces and minimal submanifolds of $S^3$……Page 295
7.6 Pinkall’s inequality……Page 296
7.7 Related work by Robert L. Bryant……Page 299
7.8 The conformal invariants of Li and Yau, and of Gromov……Page 303
7.9 Contributions from the German school and their colleagues……Page 305
7.10 Other recent contributions to the subject……Page 312
7.11 Exercises……Page 316
Appendix: Partitions of unity……Page 317
Bibliography……Page 323
Index……Page 329