Differential Geometry For Physicists And Mathematicians: Moving Frames And Differential Forms: From Euclid Past Riemann Abridged edition [Hardback]

Vargas introduces differential geometry using the method of moving frames and the exterior calculus. After introducing the basic theory of differential forms and pertinent algebra, he turns to "flat cases" known as affine and Euclidean spaces, and simple examples of their generalizations. The point is to explain advanced concepts by dealing with them first in simple structures. He covers tools, two Klein geometries, Cartan connections, and the future. Among specific topics are vector spaces and tensor products, affine Klein geometry, generalized geometry made simple, Riemannian spaces and pseudo-spaces, extensions of Cartan, and a book of farewells. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)

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.