By Jacques Lafontaine
This e-book is an advent to differential manifolds. It provides sturdy preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie idea. It presupposes little heritage: the reader is just anticipated to grasp simple differential calculus, and a bit point-set topology. The ebook covers the most themes of differential geometry: manifolds, tangent house, vector fields, differential varieties, Lie teams, and some extra subtle subject matters similar to de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.
Its ambition is to provide stable foundations. particularly, the creation of “abstract” notions equivalent to manifolds or differential kinds 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 specialist reader. recommendations are supplied for many of them.
The booklet 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 gorgeous theory.
The unique French textual content advent aux variétés différentielles has been a best-seller in its type in France for lots of years.
Jacques Lafontaine was once 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. 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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0 The result is therefore a consequence of the uniform continuity of f on K. 41. Let C be the set of critical points. It suﬃces to show that f (C ∩ A) is of measure zero for every cube A. We ﬁrst note that if x ∈ C, the vector space Im f (x) is contained in a hyperplane H ⊂ Rn . Let r > 0, and y be such that y − x < r. 38 An Introduction to Diﬀerential Manifolds By the lemma, the distance from f (y) to the aﬃne hyperplane H parallel to H and containing f (x) is less than λ(r). On the other hand, if K = supx∈B f (x) , we have f (y) − f (x) < Kr.
Suppose f is a C 1 map from an open subset U of Rp to Rq . Suppose that 0 ∈ U and that the diﬀerential df0 is surjective. Then there exists an open subset V in Rp containing 0 and a diﬀeomorphism ψ of W to its image such that ψ(W ) ⊂ U and f ψ(x1 , . . , xp ) = (x1 , . . , xq ). Proof. Necessarily p q. This time by permuting the xi coordinates, we can suppose that the matrix B = ∂j f i (0) 1 i q, 1 j q is invertible, and we deﬁne a map h from U to Rp by h(x) = f 1 (x), . . , f q (x), xq+1 , .
A map f from an open subset U of Rp to an open subset V in Rq is a C k diﬀeomorphism if it admits a C k inverse. We say that U and V are diﬀeomorphic. Denote the inverse map by g. The chain rule applied to f ◦ g and g ◦ f tells us that if a ∈ U , the linear maps dfa and dgf (a) are mutual inverses. In particular, this forces p = q. Remark. It is also true that an open subset of Rp cannot be homeomorphic to an open subset of Rq unless p = q. This result, called the invariance of domain, is distinctly more diﬃcult to prove, and appeals to algebraic topology (for a proof, see for example [Karoubi-Leruste 87, Chapter V] or [Dugundji 65, Chapter XVII, no.
An Introduction to Differential Manifolds by Jacques Lafontaine