Curvature in mathematics and physics, paperback/shlomo sternberg

This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics.

Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einsteins theory of general relativity, using the Cartan exterior calculus as a principal tool.

Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds.

A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civitas theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gausss lemma, and variational formulas.

Additional topics include the Hopf-Rinow, Myers, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types.

Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.