By Jacques Lafontaine

This booklet is an advent to differential manifolds. It supplies reliable preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie conception. It presupposes little historical past: 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 kinds, Lie teams, and some extra refined themes equivalent to de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide strong foundations. specifically, the creation of “abstract” notions akin to manifolds or differential types is encouraged 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 refined, can help the newbie in addition to the extra specialist reader. suggestions are supplied for many of them.

The publication can 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 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 collage 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 facets of mathematical relativity. along with his own learn articles, he was once serious about numerous textbooks and study monographs.

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**Example text**

Ki / Vi . Ki / Vi for all i 2 I and . x/ . Ki /: The strong topology has all possible sets of this form for a base. X; Y /. We refer the reader to [Hirsch 1976], from which we have freely borrowed. 9. Let X D T N and X 0 D T 0 N 0 be trivial foliated spaces 0 (with T Rp , T 0 Rp ). X; X 0 /. Proof. Since all functions preserve leaves we may assume that X 0 D T 0 D ޒn , regarded as a foliated space with one leaf. X; ޒn / in the strong topology. Let fV˛ g be a locally fi nite open cover of X and for each ˛ let ˛ > 0.

Assume that the isotropy group at x; fg 2 G j gx D xg, has dimension independent of x. Then M is foliated by the orbits of G. (If 1 In the fi rst edition we relied upon [Lawson 1977] as a basic reference for foliations, and this is still a valuable source. The more recent books [Candel and Conlon 2000; 2003] give an extensive and thorough treatment of the geometric and topological structure of foliated manifolds and foliated spaces. The books and their authors have been very helpful to us in the preparation of the second edition.

More generally, if V is any fi nite-dimensional subspace of H , let us choose an orthonormal basis P '1 ; : : : ; 'n for V . x/ iD1 is independent of all choices. V / can be thought of as describing how the total dimension of V is distributed or localized over the space X . x/; iD1 where the inner product is taken pointwise in Hx , is a signed measure of total mass equal to the trace of T and which again describes how this total trace is distributed over the space X . V / is the orthogonal projection onto 16 I.