Topics in Differential Geometry

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.

Inhalt

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
3. Introduction
4. Proofs of the Theorems
   Appendix
   References
   (G, E) Structures
   References
   Tensorial Concomitants of an Almost Complex Structure
5. Introduction
6. A Special Chart for an Almost Complex Structure
7. A "Natural" Hermitian, Symmetric, Bilinear Form on an Almost Complex Manifold
8. The S Derivative
   References
   Variétés Symplectiques, Variétés Canoniques, et Systèmes Dynamiques
   Introduction
   I. Variétés Symplectiques et Variétés de Contact
9. Variétés Symplectiques Exactes
10. Variétés Symplectiques Exactes et Varietes de Contact
    II. Systèmes Dynamiques
11. La Variété de Contact des États d'un Systeme Dynamique
12. Systeme Differentiel sur la Variete de Contact des Etats
13. Le Système Différentiel Usuel de Hamilton
    III. Transformations Canoniques
14. Notion de Structure Canonique
15. L'ideal I, de l'Algèbre Extérieure des Formes d'Une Variété Canonique et Les Cartes Canoniques
16. Transformations Canoniques de (W, F, t)
17. Transformations Canoniques de (W~, G~, t)
18. Cas d'Une Variété Canonique á 2-forme
19. Variétés Exactes
    Bibliographie
    Divergence-Free Third Order Concomitants of the Metric Tensor in Three Dimensions
20. Introduction
21. The Uniqueness of Hij
    References
    A Functional Equation in the Characterization of Null Cone Preserving Maps
22. Introduction
23. Basic Hypotheses
24. Reduction to Functional Equations
25. Reduction to One Unknown Function
26. Reduction to Cauchy's Equation
27. Unification of Results
28. Additional Remarks
    References
    Generalized Clebsch Representations on Manifolds
29. Introduction
30. The Generalized Clebsch Representation
31. The Gauge Transformations
32. Associated Variational Problems
33. The Case n = 3
34. The Case n = 4
35. Higher Order Variational Problems Resulting from Clebsch Representations
    References
    Note on Locally Symmetric Vector Fields in a Riemannian Space
36. Introduction
37. Symmetry
38. First Order Local Symmetry
    > 3
    > 3: Spaces of Constant Curvature
39. n = 3
40. n = 3: Spaces of Constant Curvature
41. Second Order Local Symmetry
    > 3
42. Second Order Symmetry: n = 3
43. Orientation of Galaxies
    Mean Curvature of Immersed Manifolds
46. Immersions in Riemannian Manifolds
47. Immersions of Surfaces in S3
48. Conformai Invariants
    References
    Connections and M-Tensors on the Tangent Bundle TM
49. Introduction
50. The Tangent Bundle and the Slit Tangent Bundle
51. Connections and M-Tensors and Their Simple Properties
52. (1, 1)-Connections as Horizontal Distributions on TM
53. Vector Fields on TM and Their Relation with a (1, 1)-Connection
54. (1, 0)-Connection on STM as Systems of Paths in M and as Second Order Differential Equations on M
55. Mappings Between Connections of Different Types and Their Compositions
56. Decomposition Theorems
    References
    Differential Geometry of Totally Real Submanifolds
57. Introduction
58. Preliminaries
59. Totally Real Submanifolds
60. Covariant Derivatives of fxi, fhy, and fxy
61. The Case in Which M2m Is a Complex Space Form
62. The Case in Which the Bochner Curvature Tensor of M2m Vanishes
    References