Differential Geometry for Physicists and Mathematicians

Moving Frames and Differential Forms: From Euclid Past Riemann

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