Dillen / Verstraelen Handbook of Differential Geometry

1. Auflage 2005
ISBN: 978-0-08-046120-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 574 Seiten, Web PDF

In the series of volumes which together will constitute the 'Handbook of Differential Geometry' we try to give a rather complete survey of the field of differential geometry. 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. In this second volume a wide range of areas in the very broad field of differential geometry is discussed, as there are Riemannian geometry, Lorentzian geometry, Finsler geometry, symplectic geometry, contact geometry, complex geometry, Lagrange geometry and the geometry of foliations. Although this does not cover the whole of differential geometry, the reader will be provided with an overview of some its most important areas.
. Written by experts and covering recent research
. Extensive bibliography
. Dealing with a diverse range of areas
. Starting from the basics

Autoren/Hrsg.

Dillen, Franki J. E.

Verstraelen, Leopold C. A.

Inhaltsverzeichnis

1;Cover;1
2;Contents;12
3;Preface;8
4;List of Contributors;10
5;Contents of Volume I;14
6;Some Problems on Finsler Geometry;16
6.1;Introduction;18
6.2;Preliminaries;19
6.3;Volume and area in Finsler spaces;24
6.4;Unit spheres in Minkowski spaces;27
6.5;Symplectic equivalence of Finsler manifolds;29
6.6;Around Hilbert's fourth problem;34
6.7;Closed geodesics;37
6.8;Differential invariants of Finsler surfaces;38
6.9;References;46
7;Foliations;50
7.1;Foreword;52
7.2;Definitions and examples;52
7.3;Transverse structures;61
7.4;Codimension one foliations;65
7.5;Gamma-structures;66
7.6;The leaf space;67
7.7;Characteristic classes;69
7.8;Basic global analysis;70
7.9;Deformation theory of foliations;76
7.10;Some other themes;78
7.11;References;81
8;Symplectic Geometry;94
8.1;Introduction;96
8.2;Symplectic manifolds;97
8.3;Lagrangian submanifolds;108
8.4;Complex structures;123
8.5;Symplectic geography;139
8.6;Hamiltonian geometry;154
8.7;Symplectic reduction;175
8.8;References;198
9;Metric Riemannian Geometry;204
9.1;Introduction;206
9.2;Sphere theorems;209
9.3;Finiteness theorems and Gromov-Hausdorff distance;210
9.4;Geodesic coordinate, injectivity radius, comparison theorems and sphere theorem;213
9.5;Packing and precompactness theorem;219
9.6;Construction of homeomorphism by isotopy theory;223
9.7;Harmonic coordinate and its application;225
9.8;Center of mass technique;227
9.9;Embedding Riemannian manifolds by distance function;230
9.10;Almost flat manifold;232
9.11;Collapsing Riemannian manifolds-I;235
9.12;Collapsing Riemannian manifolds-II;239
9.13;Collapsing Riemannian manifolds-III;244
9.14;Morse theory of distance function;247
9.15;Finiteness theorem by Morse theory;252
9.16;Soul theorem and splitting theorem;254
9.17;Alexandrov space-I;260
9.18;Alexandrov space-II;269
9.19;First Betti number and fundamental group;278
9.20;Hausdorff convergence of Einstein manifolds;286
9.21;Sphere theorem and L2 comparison theorem;289
9.22;Hausdorff convergence and Ricci curvature-I;297
9.23;Hausdorff convergence and Ricci curvature-II;306
9.24;References;323
10;Contact Geometry;330
10.1;Introduction;332
10.2;Contact manifolds;332
10.3;Contact structures on 3-manifolds;366
10.4;A guide to the literature;391
10.5;References;393
11;Complex Differential Geometry;398
11.1;Introduction;400
11.2;Complex manifolds;400
11.3;Almost complex structures;406
11.4;Dolbeault Lemma;408
11.5;Kaehler manifolds;410
11.6;Complex space forms;416
11.7;Laplace-Beltrami operator on a Hermitian manifold;418
11.8;Harmonic differential forms on Kaehler manifolds;422
11.9;Applications;428
11.10;Chern classes;431
11.11;Deformation of complex structures;441
11.12;References;448
12;Compendium on the Geometry of Lagrange Spaces;452
12.1;Introduction;454
12.2;Tangent bundle;455
12.3;Lagrange spaces;471
12.4;Finsler spaces;490
12.5;The geometry of T(k)M;497
12.6;Lagrange spaces of higher order;517
12.7;References;526
13;Certain Actual Topics on Modern Lorentzian Geometry;528
13.1;Preface;530
13.2;Some aspects on the topology of Lorentzian manifolds;530
13.3;Geodesics and completeness;534
13.4;Curvature of Lorentzian manifolds;537
13.5;The Bochner technique on Lorentzian manifolds;550
13.6;References;558
14;Author Index;562
15;Subject Index;570