By Jacques Lafontaine

This e-book is an creation to differential manifolds. It offers reliable preliminaries for extra complicated subject matters: Riemannian manifolds, differential topology, Lie concept. It presupposes little heritage: the reader is simply anticipated to grasp uncomplicated differential calculus, and a bit point-set topology. The publication covers the most issues of differential geometry: manifolds, tangent area, vector fields, differential kinds, Lie teams, and some extra refined issues equivalent to de Rham cohomology, measure concept and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide strong foundations. specifically, the creation of “abstract” notions comparable to manifolds or differential varieties is stimulated 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 professional reader. recommendations are supplied for many of them.

The ebook 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 creation aux variétés différentielles has been a best-seller in its class in France for lots of years.

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

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

**Example text**

An element in 7ln >X(F). 1 (Y,3Y,x) satisfies the further condition G that 3f:3X * 3Y homeomorphism). , a G diffeomorphism or More generally we define 71r codimension 0 submanifold of the boundary of Y. (Y,3 Y,X) where 3+Y is a We insist that 3f is an isomorphism on 3+X * 3+Y and a cat equivalence from 3X - 3+X to 3Y - 3+Y. The equivalence relation is the obvious modification of the one given for U° at (Y,;v). 13- 7l° at (Y,3Y,A)/~ = N°at (Y,3Y,A) «Pfl"t" An element in h JLTip (Y,X) (in 7i f\ Q + 7lc (Y,X) (Y,3Y,X)) require bundle data.

It covers f and decomposes in the same way as the spaces do. , has torsion x = x(f). The Mayer-Vietoris property implies T(7 1 ) + x(F) = 2x(f), and our claim follows. Let f :B' > B be again a G homotopy equivalence with torsion x and E ->• B a — * G vector bundle. Let f :f DE • DE be a map of the disk bundles which covers f and extends f :f SE -• SE. Then we find f :(f E) * + E*, the induced map on Thorn spaces. 4 + V e/TM (T) Vi tr S ( E ) V T ; T+ Tx(f r ) = x. Finally, let E be a G vector bundle over B and V a complex representation.

Note that these are Z[G] modules. Let $ be as above, let E be a closed subset of J{G), and H a minimal element in J{G) - E. Assume that $ is E^ good. 3 in [W3l . 6. Suppose, with A = Z[NH/H] coefficients, that Hi($H) = 0 for i < r. Then: (i) If H r + 1 U H ,B) = 0 for every right A module B, then J^U1*) is a projective A module, (ii) If (WH,XH) and (ZH,YH) are finite NH/H complexes, H r U H ) is finitely generated. TT (iii) If in addition to (i) and the conclusion of (ii) H^($ ) = 0 for i f- r, then H r U H ) is stably free.