Topics in Differential Geometry

List of Contributors

Preface

Evan Tom Davies

References

Reminiscences of E. T. Davies


1. Projective Differential Geometry


2. Theory of Connections

References

The Uniqueness of the Neutrino Energy-Momentum Tensor and the Einstein-Weyl Equations


1. Introduction


2. Proofs of the Theorems

Appendix

References

(G, E) Structures

References

Tensorial Concomitants of an Almost Complex Structure


1. Introduction


2. A Special Chart for an Almost Complex Structure


3. A "Natural" Hermitian, Symmetric, Bilinear Form on an Almost Complex Manifold


4. The S Derivative

References

Variétés Symplectiques, Variétés Canoniques, et Systèmes Dynamiques

Introduction

I. Variétés Symplectiques et Variétés de Contact


1. Variétés Symplectiques Exactes


2. Variétés Symplectiques Exactes et Varietes de Contact

II. Systèmes Dynamiques


3. La Variété de Contact des États d'un Systeme Dynamique


4. Systeme Differentiel sur la Variete de Contact des Etats


5. Le Système Différentiel Usuel de Hamilton

III. Transformations Canoniques


6. Notion de Structure Canonique


7. L'ideal I, de l'Algèbre Extérieure des Formes d'Une Variété Canonique et Les Cartes Canoniques


8. Transformations Canoniques de (W, F, t)


9. Transformations Canoniques de (W˜, G˜, t)


10. Cas d'Une Variété Canonique á 2-forme


11. Variétés Exactes

Bibliographie

Divergence-Free Third Order Concomitants of the Metric Tensor in Three Dimensions


1. Introduction


2. The Uniqueness of Hij

References

A Functional Equation in the Characterization of Null Cone Preserving Maps


1. Introduction


2. Basic Hypotheses


3. Reduction to Functional Equations


4. Reduction to One Unknown Function


5. Reduction to Cauchy's Equation


6. Unification of Results


7. Additional Remarks

References

Generalized Clebsch Representations on Manifolds


1. Introduction


2. The Generalized Clebsch Representation


3. The Gauge Transformations


4. Associated Variational Problems


5. The Case n = 3


6. The Case n = 4


7. Higher Order Variational Problems Resulting from Clebsch Representations

References

Note on Locally Symmetric Vector Fields in a Riemannian Space


1. Introduction


2. Symmetry


3. First Order Local Symmetry


4. n > 3


5. n > 3: Spaces of Constant Curvature


6. n = 3


7. n = 3: Spaces of Constant Curvature


8. Second Order Local Symmetry


9. Second Order Symmetry: n > 3


10. Second Order Symmetry: n = 3


11. Orientation of Galaxies

Mean Curvature of Immersed Manifolds


1.


2.


3. Immersions in Riemannian Manifolds


4. Immersions of Surfaces in S3


5. Conformai Invariants

References

Connections and M-Tensors on the Tangent Bundle TM


1. Introduction


2. The Tangent Bundle and the Slit Tangent Bundle


3. Connections and M-Tensors and Their Simple Properties


4. (1, 1)-Connections as Horizontal Distributions on TM


5. Vector Fields on TM and Their Relation with a (1, 1)-Connection


6. (1, 0)-Connection on STM as Systems of Paths in M and as Second Order Differential Equations on M


7. Mappings Between Connections of Different Types and Their Compositions


8. Decomposition Theorems

References

Differential Geometry of Totally Real Submanifolds


0. Introduction


1. Preliminaries


2. Totally Real Submanifolds


3. Covariant Derivatives of fxi, fhy, and fxy


4. The Case in Which M2m Is a Complex Space Form


5. The Case in Which the Bochner Curvature Tensor of M2m Vanishes

References