By Jacques Lafontaine

ISBN-10: 3319207350

ISBN-13: 9783319207353

This publication is an creation to differential manifolds. It offers good preliminaries for extra complicated issues: Riemannian manifolds, differential topology, Lie conception. It presupposes little history: the reader is simply anticipated to grasp simple differential calculus, and a bit point-set topology. The publication covers the most themes of differential geometry: manifolds, tangent house, vector fields, differential types, Lie teams, and some extra refined issues corresponding to de Rham cohomology, measure concept and the Gauss-Bonnet theorem for surfaces.

Its ambition is to offer strong foundations. specifically, the advent of “abstract” notions akin to manifolds or differential varieties is stimulated through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty routines, a few of them effortless and classical, a few others extra refined, may help the newbie in addition to the extra professional reader. options are supplied for many of them.

The e-book might be of curiosity to varied readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to collect a few feeling approximately this gorgeous theory.

The unique French textual content creation aux variétés différentielles has been a best-seller in its type in France for lots of years.

Jacques Lafontaine used to be successively assistant Professor at Paris Diderot college and Professor on the college of Montpellier, the place he's shortly emeritus. His major study pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. along with his own examine articles, he used to be all in favour of a number of textbooks and examine monographs.

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**Extra resources for An Introduction to Differential Manifolds**

**Sample text**

The following result is obvious. 7) L E M M A . Let SP be a proximity base of X and let 7 c: X. Then 3PY = [U'nT:ZJ'

7) D E F I N I T I O N . A topological space with a boundedness 38 is compactly bounded iff £8 = {B c= X:Bis compact). 8) T H E O R E M . A Hausdorff space X is locally compact if and only if X being compactly bounded implies X is locally bounded. 9) DEFINITION. X is boundedly compact (' MonteV space) iff every closed bounded subset of X is compact. In proving the following theorem and corollary, we use two well-known characterizations of compactness: A subset E of a topological space X is compact iff every filter base in E has a cluster point in E iff every ultrafilter in E converges to a point in E.

Local proximity spaces In Section 7 it was shown that a compact Hausdorff space 3C is (uniquely) determined by a dense proximity space (X, 8), and that A8B in X iff A 0 B + 0 where the closures are taken in 2€. It will be shown in this section that a locally compact Hausdorff COMPACTIFICATIONS OF PROXIMITY SPACES 53 space Y is determined by a dense subspace X when we know not only the proximity of X, but also which sets in X have compact closures in Y. ) Boundedness in topological spaces has been studied axiomatically by various authors since 1939.

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