Curvature in mathematics and physics /

Curvature in mathematics and physics /

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Bibliographic Details
Author / Creator:	Sternberg, Shlomo.
Imprint:	Mineola, New York : Dover Publications, Inc., 2012.
Description:	405 pages : portrait ; 24 cm
Language:	English
Series:	Dover books on mathematics Dover books on mathematics.
Subject:	Curvature. Semi-Riemannian geometry. Geometry, Differential. Differential calculus. Algebras, Linear. Relativity (Physics) Algebras, Linear. Curvature. Differential calculus. Geometry, Differential. Relativity (Physics) Semi-Riemannian geometry.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/13552353

Hidden Bibliographic Details
ISBN:	9780486478555 0486478556
Notes:	Includes bibliographical references and index.
Summary:	"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."--Back cover.

Table of Contents:

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