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251 pages, yr 2003, revised edition, 1Mb PDF file

Abstract:

An interesting book on Semi-Riemannian Geometry and General Relativity written by an established mathematician and mathematical physicist. Concise author's quote is as follows: "Chapter I introduces the various curvatures associated to a hypersurface embedded in Euclidean space, motivated by the formula for the volume for the region obtained by thickening the hypersurface on one side... The precise definitions are given in the text. The first chapter culminates with Gauss's Theorema egregium which asserts that if we thicken a two dimensional surface evenly on both sides, then the integrands depend only on the intrinsic geometry of the surface, and not on how the surface is embedded." Of special importance are the superalgebras treated in Chapter 2, as shown in the Ch. 2 Contents: 2 Rules of calculus. 31 2.1 Superalgebras. . . . 33 2.5 Pullback. . . . . . . . . 34 2.6 Chain rule. . . . . . . . . .35 2.7 Lie derivative. . . . . . 35 2.8 Weil's formula. . . . . . . 36 2.9 Integration. . . . . . . 38 2.10 Stokes theorem. . . . . . . . . 38 2.11 Lie derivatives of vector fields. . . . . . . . . 39 2.12 Jacobi's 40 2.13 Left invariant forms. . . . . . . . . . . 41 2.14 The Maurer Cartan equations. . 43 2.15 Restriction to a subgroup . . . . .43 2.16 Frames. . . . . . . . . . . 44 2.17 Euclidean 45 2.18 Frames adapted to a submanifold. . . . 47 2.19 Curves and surfaces - their structure equations. . . 48 2.20 The sphere as an example. . . . . 48 2.21 Ribbons . . . . . . . . . . . 50 2.22 Developing a ribbon. . . . . . . . . . 51 2.23 Parallel transport along a ribbon. . . 52 \\\ "Chapter VIII is the high point of the course from the theoretical point of view. We discuss Einstein's general theory of relativity from the point of view of the Einstein--Hilbert functional. In fact we borrow the title of Hilbert's paper for the Chapter heading. We also introduce the principle of general covariance, first introduced by Einstein, Infeld, and Hoffmann to derive the 'geodesic principle' and give a whole series of other applications of this principle." ==== Free download on line at:== http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf

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http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf

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