Differential Geometry For Physicists And Mathematicians: Moving Frames And Differential Forms: From Euclid Past Riemann

Moving Frames and Differential Forms: From Euclid Past Riemann

José G Vargas

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814566414
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative — almost like a story being told — that does not impede sophistication and deep results.

It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter.

In an appendix on the classical theory of curves and surfaces, the author slashes not only the main proofs of the traditional approach, which uses vector calculus, but even existing treatments that also use differential forms for the same purpose.

Contents:

• Introduction:
  • Orientations
• Tools:
  • Differential Forms
  • Vector Spaces and Tensor Products
  • Exterior Differentiation
• Two Klein Geometries:
  • Affine Klein Geometry
  • Euclidean Klein Geometry
• Cartan Connections:
  • Generalized Geometry Made Simple
  • Affine Connections
  • Euclidean Connections
  • Riemannian Spaces and Pseudo-Spaces
• The Future?:
  • Extensions of Cartan
  • Understand the Past to Imagine the Future
  • A Book of Farewells

Readership: Physicists and mathematicians working on differential geometry.
Key Features:

• Reader Friendly
• Naturalness
• Respect for the history of the subject and related incisiveness

Part IV

CARTAN CONNECTIONS

Chapter 7

GENERALIZED GEOMETRY MADE SIMPLE

7.1 Of connections and topology

This is a transitional chapter in the sense that we study simple examples of Cartan’s generalization of affine and Euclidean Klein geometries. For further perspective, we also devote a section to the original Riemannian geometry, where Riemann’s curvature had to do with a so-called problem and method of equivalence. “Equivalence” is not a topic of differential geometry proper, but may be used there nevertheless.

We shall study three common surfaces under two different rules to compare tangent vectors at different points on those surfaces. One of these rules is the so called Levi-Civita connection (LCC). Formulated in 1917, it was the first known affine connection (we shall later call it Euclidean rather than affine). It was adopted, may be unnecessarily, by general relativity. We say “may be unnecessarily” because eventually there were other connections compatible with the original Riemannian geometry and with general relativity, but with better properties.

The most obvious alternative to the LCC is the Columbus connection (see Preface), the reason for this choice of name later to become clear. It is of the type called teleparallel, which means that one can then establish a relation of geometric equality of tangent vectors at different points of a region of a manifold. If there are exceptional points (even as few as just one) where we cannot define the equality of its tangent vectors to the tangent vectors at other points of M, we say that the teleparallelism is local or limited to regions of the manifold. The topology of the manifold may impede defining the connection everywhere, while allowing it on regions.

Consider the following easy-to-understand example of a local property. The direction East (or any other direction for that matter) is not defined at the poles where, correspondingly, the Columbus connection is not defined either. One cannot put connections other than the LCC on the full 2-sphere, i.e. the sphere in 3-dimensional Euclidean space. We then say that they are not globally defined. It is a topological issue.

The Columbus connection is globally defined on manifolds that are regions of a 2-sphere, among them the one resulting from puncturing it at the poles. Puncturing changes the topology of the sphere. This difference is largely academic for the physicist who integrates his equation to find the world in which he lives. Cartan indeed spoke of systems of equations whose solutions are the manifolds in which the connection lives [29]. Thus the 2-sphere would be the solution to a system of equations of structure with the LCC, but not with any other connection.

Assume that, making abstraction of dimensionality and of signature of the metric, the spacetime solution of some hypothetical system of equations were like a sphere but with Columbus connection. It would not be the whole sphere but just a region of it. The energy required to create the region would be greater and greater as one approached the full sphere. Only infinite energy would close the surface, since it is to be expected that the singularity of the torsion at the poles would entail a singularity of required energy, as well as singularity of other physical quantities related to affine (said better, Euclidean) structure. Hence, we shall take the concept of teleparallelism and zero affine curvature as synonymous, at least from a physical perspective.

...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Acknowledgements
• Preface
• Contents
• I INTRODUCTION
• II TOOLS
• III TWO KLEIN GEOMETRIES
• IV CARTAN CONNECTIONS
• V THE FUTURE?
• APPENDIX A: GEOMETRY OF CURVES AND SURFACES
• APPENDIX B: “BIOGRAPHIES” (“PUBLI ”GRAPHIES)
• APPENDIX C: PUBLICATIONS BY THE AUTHOR
• References
• Index