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Curvature in Mathematics and Physics (Dover Books on Mathematics)by Shlomo Sternberg
Synopses & ReviewsPublisher Comments: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 worldrenowned mathematician for more than fifty years, the treatment introduces semiRiemannian 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 LeviCivita's theorem, biinvariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the HopfRinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semiRiemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms. Synopsis: Expert treatment introduces semiRiemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.
Table of ContentsIntroduction 1. Gauss's Theorem Egregium 2. Rules of Calculus 3. Connections on the Tangent Bundle 4. LeviCivita's Theorem 5. Biinvariant Metrics on a Lie Group 6. Cartan Calculations 7. Gauss's Lemma 8. Variational Formulas 9. The HopfRinow 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. SemiRiemannian Submersions Bibliography Index What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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