By Theodore Frankel

ISBN-10: 0521387531

ISBN-13: 9780521387538

This publication is meant to supply a operating wisdom of these elements of external differential types, differential geometry, algebraic and differential topology, Lie teams, vector bundles and Chern varieties which are crucial for a deeper figuring out of either classical and smooth physics and engineering. integrated are discussions of analytical and fluid dynamics, electromagnetism, thermodynamics, the deformation tensors of elasticity, cleaning soap movies, targeted and normal relativity, the Dirac operator and spinors, and gauge fields, together with Yang-Mills, the Aharonov-Bohm influence, Berry part, and instanton winding numbers. earlier than discussing summary notions of differential geometry, geometric instinct is built via a slightly vast creation to the learn of surfaces in usual area; as a result, the booklet must also be of curiosity to arithmetic scholars. This e-book might be helpful to graduate and complex undergraduate scholars of physics, engineering and arithmetic. it may be used as a path textual content or for self research.

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I. Two-Dimensional Manifolds of Bounded Curvature 65 Let K, and K, be rectifiable simple closed curves in the metric spaces M, and M,, and cpa map of K, onto K,. We shall say that cpmaps K, onto K, with the lengths of arcs preserved if cpmaps K i onto K, one-to-one and takes any arc of K, into an arc of K, of the same length. Suppose we are given metric spaces (M, , pi) and (M,, p2). 1. (Reshetnyak (1961a)). Then there is a convex cone Q and a map cp: Q -+ R such that the following conditions are satisfied: 1) the curvature of Q doesnot exceed w’(R); 2) cpis a contracting map; 3) the boundary of the cone Q is transformed by the map cpinto the boundary of R with the lengths of arcs preserved.

For the case when X is a boundary point of M the constructions are similar. Manifolds of Bounded Curvature 51 The proof of the theorem reduces to a consideration of neighbourhoods of points of M that arise as a result of pasting together boundary points of the manifolds M,. Suppose, say, that a point X is obtained by pasting together points Xi E M,i, i = 1, 2, . . , m. In M,, the point Xi has a neighbourhood isometric to some circular sector A(&, h). By pasting together these neighbourhoods we obtain a neighbourhood of X.

The given theorem establishes a necessary condition that is satisfied by any two-dimensional manifold of bounded curvature. This condition is also suflicient. In the part that touches on the sufficiency, the conditions imposed on the sequence of polyhedral metrics pn can be weakened. Namely, the following theorem, which we shall call the second theorem on approximation, is true. 2.

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The Geometry of Physics: An Introduction by Theodore Frankel

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