Rund / Forbes Topics in Differential Geometry

1. Auflage 2014
ISBN: 978-1-4832-7269-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 196 Seiten, Web PDF

Topics in Differential Geometry is a collection of papers related to the work of Evan Tom Davies in differential geometry. Some papers discuss projective differential geometry, the neutrino energy-momentum tensor, and the divergence-free third order concomitants of the metric tensor in three dimensions. Other papers explain generalized Clebsch representations on manifolds, locally symmetric vector fields in a Riemannian space, mean curvature of immersed manifolds, and differential geometry of totally real submanifolds. One paper considers the symmetry of the first and second order for a vector field in a Riemannnian space to arrive at conditions the vector field satisfies. Another paper examines the concept of a smooth manifold-tensor and the three types of connections on the tangent bundle TM, their properties, and their inter-relationships. The paper explains some clarification on the relationship between several related known concepts in the differential geometry of TM, such as the system of general paths of Douglas, the nonlinear connections of Barthel, ano and Ishihara, as well as the nonhomogeneous connection of Grifone. The collection is suitable for mathematicians, geometricians, physicists, and academicians interested in differential geometry.

Autoren/Hrsg.

Rund, Hanno

Forbes, William P.

Inhaltsverzeichnis

1;Front Cover;1
2;Topics in Differential Geometry;4
3;Copyright Page;5
4;Table of Contents ;6
5;List of Contributors;10
6;Preface;12
7;Chapter 1. Evan Tom Davies;14
7.1;References;19
8;Chapter 2. Reminiscences of E. T. Davies;22
8.1;1. Prqjective Differential Geometry;23
8.2;2. Theory of Connections;26
8.3;References;26
9;Chapter 3. The Uniqueness of the Neutrino Energy-Momentum Tensor and the Einstein-Weyl Equations;28
9.1;1. Introduction;28
9.2;2. Proofs of the Theorems;32
9.3;Appendix;37
9.4;Acknowledgments;40
9.5;References;40
10;Chapter 4. (G, E) Structures;42
10.1;Introduction;42
10.2;1.;42
10.3;2.;44
10.4;3.;46
10.5;4.;47
10.6;5.;49
10.7;6.;51
10.8;7.;54
10.9;8.;55
10.10;References;56
11;Chapter 5. Tensorial Concomitants of an Almost Complex Structure;58
11.1;1. Introduction;58
11.2;2. A Special Chart for an Almost Complex Structure;60
11.3;3. A "Natural" Hermitian, Symmetric, Bilinear Form on an Almost Complex Manifold;63
11.4;4. The S Derivative;66
11.5;Acknowledgments;68
11.6;References;68
12;Chapter 6. Variétés Symplectiques, Variétés Canoniques, et Systèmes Dynamiques;70
12.1;Introduction;70
12.2;1. Variétés Symplectiques Exactes;71
12.3;2. Variétés Symplectiques Exactes et Variétés de Contact;73
12.4;3. La Variété de Contact des États d'un Système Dynamique;76
12.5;4. Systéme Différentiel sur la Variété de Contact des États;78
12.6;5. Le Systeme Differentiel Usuel de Hamilton;80
12.7;6. Notion de Structure Canonique;82
12.8;7. L'idéal l , de I' Algèbre Extérieure des Formes d'Une Variété Canonique et Les Cartes Canoniques;85
12.9;8. Transformations Canoniques de (W, F, t);88
12.10;9. Transformations Canoniques de (lW, G, t);92
12.11;10. Cas d'Une Variété Canonique à 2-forme ;93
12.12;11. Variétés Exactes;95
12.13;Bibliographie;97
13;Chapter 7. Divergence-Free Third Order Concomitants of the Metric Tensor in Three Dimensions;100
13.1;1. Introduction;100
13.2;2. The Uniqueness of Hij;102
13.3;Acknowledgments;111
13.4;References;111
14;Chapter 8. A Functional Equation in the Characterization of Null Cone Preserving Maps;112
14.1;1. Introduction;112
14.2;2. Basic Hypotheses;113
14.3;3. Reduction to Functional Equations;114
14.4;4. Reduction to One Unknown Function;117
14.5;5. Reduction to Cauchy's Equation;119
14.6;6. Unification of Results;120
14.7;7. Additional Remarks;122
14.8;References;122
15;Chapter 9. Generalized Clebsch Representations on Manifolds;124
15.1;1. Introduction;124
15.2;2. The Generalized Clebsch Representation;126
15.3;3. The Gauge Transformations;131
15.4;4. Associated Variational Problems;132
15.5;5. The Case n = 3;134
15.6;6. The Case n = 4;139
15.7;7. Higher Order Variational Problems Resulting from Clebsch Representations;143
15.8;Acknowledgments;146
15.9;References;146
16;Chapter 10. Note on Locally Symmetric Vector Fields in a Riemannian Space;148
16.1;1. Introduction;148
16.2;2. Symmetry;149
16.3;3. First Order Local Symmetry;150
16.4;4. n > 3;151
16.5;5. n > 3: Spaces of Constant Curvature;153
16.6;6. n = 3;154
16.7;7. n = 3: Spaces of Constant Curvature;155
16.8;8. Second Order Local Symmetry;156
16.9;9. Second Order Symmetry: n > 3;157
16.10;10. Second Order Symmetry: n = 3;158
16.11;11. Orientation of Galaxies;158
17;Chapter 11. Mean Curvature of Immersed Manifolds;162
17.1;1.;162
17.2;2.;163
17.3;3. Immersions in Riemannian Manifolds;165
17.4;4. Immersions of Surfaces in S3;166
17.5;5. Conformai Invariants;167
17.6;References;169
18;Chapter 12. Connections and M-Tensors on the Tangent Bundle TM;170
18.1;1. Introduction;170
18.2;2. The Tangent Bundle and the Slit Tangent Bundle;171
18.3;3. Connections and M-Tensors and Their Simple Properties;173
18.4;4. (1, 1)-Connections as Horizontal Distributions on
TM;175
18.5;5. Vector Fields on TM and Their Relation with a
(1, 1)-Connection;176
18.6;6. (1, 0)-Connection on STM as Systems of Paths in M and as Second Order Differential Equations on M;178
18.7;7. Mappings between Connections of Different Types and Their Compositions;179
18.8;8. Decomposition Theorems;183
18.9;Acknowledgments;185
18.10;References;185
19;Chapter 13. Differential Geometry of Totally Real Submanifolds;186
19.1;0. Introduction;186
19.2;1. Preliminaries;187
19.3;2. Totally Real Submanifolds;189
19.4;3. Covariant Derivatives of fxi,fhy, and fxy;190
19.5;4. The Case in Which M2m Is a Complex Space Form;192
19.6;5. The Case in Which the Bochner Curvature Tensor of M2m Vanishes;194
19.7;References;197