Handbook of Differential Geometry

Franki J.E. Dillen, Leopold C.A. Verstraelen044452052X, 9780444520524, 9780080461205

In the series of volumes which together will constitute the Handbook of Differential Geometry a rather complete survey of the field of differential geometry is given. The different chapters will both deal with the basic material of differential geometry and with research results (old and recent). All chapters are written by experts in the area and contain a large bibliography.

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