Handbook of Differential Geometry (eBook)

Cover 1
Contents 12
Preface 8
List of Contributors 10
Contents of Volume I 14
Some Problems on Finsler Geometry 16
Introduction 18
Preliminaries 19
Volume and area in Finsler spaces 24
Unit spheres in Minkowski spaces 27
Symplectic equivalence of Finsler manifolds 29
Around Hilbert's fourth problem 34
Closed geodesics 37
Differential invariants of Finsler surfaces 38
References 46
Foliations 50
Foreword 52
Definitions and examples 52
Transverse structures 61
Codimension one foliations 65
Gamma-structures 66
The leaf space 67
Characteristic classes 69
Basic global analysis 70
Deformation theory of foliations 76
Some other themes 78
References 81
Symplectic Geometry 94
Introduction 96
Symplectic manifolds 97
Lagrangian submanifolds 108
Complex structures 123
Symplectic geography 139
Hamiltonian geometry 154
Symplectic reduction 175
References 198
Metric Riemannian Geometry 204
Introduction 206
Sphere theorems 209
Finiteness theorems and Gromov-Hausdorff distance 210
Geodesic coordinate, injectivity radius, comparison theorems and sphere theorem 213
Packing and precompactness theorem 219
Construction of homeomorphism by isotopy theory 223
Harmonic coordinate and its application 225
Center of mass technique 227
Embedding Riemannian manifolds by distance function 230
Almost flat manifold 232
Collapsing Riemannian manifolds-I 235
Collapsing Riemannian manifolds-II 239
Collapsing Riemannian manifolds-III 244
Morse theory of distance function 247
Finiteness theorem by Morse theory 252
Soul theorem and splitting theorem 254
Alexandrov space-I 260
Alexandrov space-II 269
First Betti number and fundamental group 278
Hausdorff convergence of Einstein manifolds 286
Sphere theorem and L2 comparison theorem 289
Hausdorff convergence and Ricci curvature-I 297
Hausdorff convergence and Ricci curvature-II 306
References 323
Contact Geometry 330
Introduction 332
Contact manifolds 332
Contact structures on 3-manifolds 366
A guide to the literature 391
References 393
Complex Differential Geometry 398
Introduction 400
Complex manifolds 400
Almost complex structures 406
Dolbeault Lemma 408
Kaehler manifolds 410
Complex space forms 416
Laplace-Beltrami operator on a Hermitian manifold 418
Harmonic differential forms on Kaehler manifolds 422
Applications 428
Chern classes 431
Deformation of complex structures 441
References 448
Compendium on the Geometry of Lagrange Spaces 452
Introduction 454
Tangent bundle 455
Lagrange spaces 471
Finsler spaces 490
The geometry of T(k)M 497
Lagrange spaces of higher order 517
References 526
Certain Actual Topics on Modern Lorentzian Geometry 528
Preface 530
Some aspects on the topology of Lorentzian manifolds 530
Geodesics and completeness 534
Curvature of Lorentzian manifolds 537
The Bochner technique on Lorentzian manifolds 550
References 558
Author Index 562
Subject Index 570