An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This e-book is an creation to differential manifolds. It provides stable preliminaries for extra complex subject matters: Riemannian manifolds, differential topology, Lie thought. It presupposes little heritage: the reader is barely anticipated to grasp easy differential calculus, and a bit point-set topology. The e-book covers the most subject matters of differential geometry: manifolds, tangent area, vector fields, differential types, Lie teams, and some extra refined themes similar to de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide reliable foundations. particularly, the advent of “abstract” notions resembling manifolds or differential varieties is prompted through questions and examples from arithmetic or theoretical physics. greater than a hundred and fifty routines, a few of them effortless and classical, a few others extra subtle, can assist the newbie in addition to the extra professional reader. recommendations are supplied for many of them.

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

The unique French textual content advent aux variétés différentielles has been a best-seller in its class 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 almost immediately emeritus. His major learn pursuits are Riemannian and pseudo-Riemannian geometry, together with a few features of mathematical relativity. in addition to his own examine articles, he was once focused on a number of textbooks and examine monographs.

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Weakening of the regularity hypotheses In this book we essentially work with smooth functions. It is not an easy task to relieve ourselves of this hypothesis. 42 An Introduction to Differential Manifolds It turns out that the space of smooth functions cannot be equipped with a norm (which takes into account the convergence of all derivatives). Thus they do not easily lend themselves to functional analysis. In the majority of problems of analysis on domains of Rn or manifolds, we frequently work with L2 spaces and Sobolev spaces (for example the space of f such that f and df are L2 ), and then prove regularity theorems.

A symmetric endomorphism S is said to be positive if Sx, x 0 for all x, and strictly positive if Sx, x > 0 for all x = 0. 44 An Introduction to Differential Manifolds a) Show that if S is strictly positive, then S is invertible; show that a symmetric endomorphism is strictly positive if and only if there exists a real number k > 0 such that ∀x ∈ Rn , Sx, x k x 2 (use diagonalization for symmetric matrices). Deduce that the set of n(n+1) strictly positive endomorphisms is an open subset of R 2 . b) Show that every strictly positive endomorphism S has a unique strictly positive square root, and the map S −→ T is a diffeomorphism.

A C k immersion from an open subset U ⊂ Rp to Rq is C k map from U to Rq with injective differential at each point. A C k submersion is a C k map from U in Rq with surjective differential at each point. With this notation, we note that p q if f is an immersion, and p q if f is a submersion. Of course a map that is both an immersion and submersion is a local diffeomorphism. Remarks a) If the differential at point a is injective (resp. surjective) there exists an open subset containing a for which this property subsists.

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