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, Einstein's 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-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, 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.
By:
Shlomo Sternberg, Mathematics Imprint: Dover Publications Inc. Country of Publication: United States Dimensions:
Height: 234mm,
Width: 158mm,
Spine: 22mm
Weight: 597g ISBN:9780486478555 ISBN 10: 0486478556 Series:Dover Books on Mathema 1.4tics Pages: 416 Publication Date:19 September 2012 Audience:
College/higher education
,
A / AS level
Format:Paperback Publisher's Status: Unspecified
Introduction 1. Gauss's Theorem Egregium 2. Rules of Calculus 3. Connections on the Tangent Bundle 4. Levi-Civita's Theorem 5. Bi-invariant Metrics on a Lie Group 6. Cartan Calculations 7. Gauss's Lemma 8. Variational Formulas 9. The Hopf-Rinow Theorem 10. Curvature, Distance and Volume 11.Review of Special Relativity 12. The Star Operator and Electromagnetism 13. Preliminaries to the Einstein Equation 14. Die Grundlagen der Physik 15. The Frobenius Theorem 16. Connections on Principal Bundles 17. Reduction of Principal Bundles 18. Superconnections 19. Semi-Riemannian Submersions Bibliography Index
