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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. An appendix on the classic theory of curves and surfaces slashes is included. It does not only contain the traditional approach that uses vector calculus, but also the treatments of the subject by those who have already used differential forms for the same purpose.
List of 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.
