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Curvature in Mathematics and Physics (Dover Books on Mathematics)

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Curvature in Mathematics and Physics (Dover Books on Mathematics) Cover

 

Synopses & Reviews

Publisher 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 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.

Synopsis:

Expert 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. Prerequisites include linear algebra and advanced calculus. 2012 edition.

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 
 

Product Details

ISBN:
9780486478555
Author:
Sternberg, Shlomo
Publisher:
Dover Publications
Author:
Mathematics
Subject:
Geometry - Differential
Subject:
semi-Riemannian
Subject:
Differential geometry
Subject:
Cartan exterior calculus
Subject:
Mathematics-Differential Geometry
Copyright:
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
20120931
Binding:
TRADE PAPER
Language:
English
Pages:
416
Dimensions:
9.25 x 6.13 in

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Related Subjects


History and Social Science » World History » General
Science and Mathematics » Mathematics » Advanced
Science and Mathematics » Mathematics » Differential Geometry
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry
Science and Mathematics » Physics » General
Science and Mathematics » Physics » Math
Science and Mathematics » Physics » Relativity Theory

Curvature in Mathematics and Physics (Dover Books on Mathematics) New Trade Paper
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Product details 416 pages Dover Publications - English 9780486478555 Reviews:
"Synopsis" by ,
Expert 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. Prerequisites include linear algebra and advanced calculus. 2012 edition.
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