Topics in Differential Geometry

Topics in Differential Geometry

1st Edition - January 1, 1976

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  • Editors: Hanno Rund, William F. Forbes
  • eBook ISBN: 9781483272696

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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.

Table of Contents

  • List of Contributors


    Evan Tom Davies


    Reminiscences of E. T. Davies

    1. Projective Differential Geometry

    2. Theory of Connections


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

    1. Introduction

    2. Proofs of the Theorems



    (G, E) Structures


    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


    Variétés Symplectiques, Variétés Canoniques, et Systèmes Dynamiques


    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


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

    1. Introduction

    2. The Uniqueness of Hij


    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


    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


    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



    3. Immersions in Riemannian Manifolds

    4. Immersions of Surfaces in S3

    5. Conformai Invariants


    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


    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


Product details

  • No. of pages: 196
  • Language: English
  • Copyright: © Academic Press 1976
  • Published: January 1, 1976
  • Imprint: Academic Press
  • eBook ISBN: 9781483272696

About the Editors

Hanno Rund

William F. Forbes

William Forbes is a Teaching Associate at Queen Mary University of London. Forbes has researched and taught upon behavioural finance for nearly twenty years. Previously, he has worked in Exeter, Manchester, Glasgow and Loughborough Universities. He is the author of Behavioural Finance (John Wiley & Son, 2009), and co-author of Corporate Governance in the United Kingdom: Past, Present and Future (Springer, 2014).

Affiliations and Expertise

Waterford Institute of Technology, Ireland and Groningen University, The Netherlands

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