Curvature in Mathematics and Physics 1st Edition by Shlomo Sternberg 9780486478555 0486478556

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Product details:

ISBN 10: 0486478556

ISBN 13: 9780486478555

Author: 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, 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.

Table of contents:

1. Gauss’s theorema 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 equations

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

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Tags: Shlomo Sternberg, Curvature, Mathematics, Physics