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:OrientationsTools:Differential FormsVector Spaces and Tensor ProductsExterior DifferentiationTwo Klein Geometries:Affine Klein GeometryEuclidean Klein GeometryCartan Connections:Generalized Geometry Made SimpleAffine ConnectionsEuclidean ConnectionsRiemannian Spaces and Pseudo-SpacesThe Future?:Extensions of CartanUnderstand the Past to Imagine the FutureA Book of Farewells
Readership: Physicists and mathematicians working on differential geometry.
Key Features:Reader FriendlyNaturalnessRespect for the history of the subject and related incisiveness