By Jacques Lafontaine

This booklet is an advent to differential manifolds. It offers strong preliminaries for extra complex subject matters: Riemannian manifolds, differential topology, Lie idea. It presupposes little history: the reader is barely anticipated to grasp uncomplicated differential calculus, and a bit point-set topology. The ebook covers the most subject matters of differential geometry: manifolds, tangent area, vector fields, differential kinds, Lie teams, and some extra subtle subject matters akin to de Rham cohomology, measure concept and the Gauss-Bonnet theorem for surfaces.

Its ambition is to offer strong foundations. particularly, the advent of “abstract” notions comparable to manifolds or differential types is inspired 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, can assist the newbie in addition to the extra specialist reader. options are supplied for many of them.

The ebook will 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 gather a few feeling approximately this pretty theory.

The unique French textual content advent aux variétés différentielles has been a best-seller in its classification in France for plenty of years.

Jacques Lafontaine used to be successively assistant Professor at Paris Diderot college and Professor on the collage of Montpellier, the place he's almost immediately emeritus. His major examine 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 excited by a number of textbooks and learn monographs.

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Furthermore, given the choice of U , we have f c 2 = exp B , with B ∈ U. Then exp 2B = f (c) = exp B, while the endomorphisms 2B and B are both in the open subset 2U , on which exp is a diﬀeomorphism, and in particular an injection. Therefore B = B/2. 1. This elegant argument was communicated to me by Max Karoubi on the terrace of a Parisian café. 32 An Introduction to Diﬀerential Manifolds The same reasoning proves that for every integer p, exp B =f 2p c . 2p Therefore by the algebraic properties of f , we have exp kB =f 2p kc 2p for all integers k and p.

On the other hand we have t diﬀeomorphisms t → et from (0, ∞) to R and t → 1−t 2 from (−1, 1) to R (for example). b) All open balls in Rn (under the Euclidean norm) are diﬀeomorphic to Rn . Using a), we see that x x −→ 1− x 2 is a diﬀeomorphism of the open ball B(0, 1) to Rn . c) In R2 , the interior of a square is diﬀeomorphic to an open disk. It suﬃces to remark that the map (x, y) −→ x y , 2 1 − x 1 − y2 is a diﬀeomorphism of the square (−1, 1) × (−1, 1) to R2 . Of course there are analogous statements in every dimension.

Y q−p = f 1 (x), . . , f p (x), y 1+f p+1 (x), . . , y q−p+f q (x) . The Jacobian matrix of g is of the form A 0 . I Chapter 1 – Diﬀerential Calculus 19 This is invertible, therefore there exists an open subset W containing 0 such that g|W is a diﬀeomorphism to its image. The required diﬀeomorphism is ϕ = g −1. Remark. An immediate consequence of this theorem is the existence of a local left inverse of f , this is to say a map from an open subset of Rq containing 0 to an open subset of Rp such that f1 ◦ f = Id Rp : it suﬃces to take f1 = (ϕ1 , .