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 concept. It presupposes little heritage: the reader is simply anticipated to grasp simple differential calculus, and a bit point-set topology. The e-book covers the most subject matters of differential geometry: manifolds, tangent house, vector fields, differential types, Lie teams, and some extra refined subject matters akin to de Rham cohomology, measure conception and the Gauss-Bonnet theorem for surfaces.
Its ambition is to offer good foundations. specifically, the advent of “abstract” notions equivalent to manifolds or differential varieties is inspired through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty workouts, a few of them effortless and classical, a few others extra refined, may help the newbie in addition to the extra specialist reader. recommendations are supplied for many of them.
The publication 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 collect a few feeling approximately this gorgeous 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 collage of Montpellier, the place he's almost immediately emeritus. His major examine pursuits are Riemannian and pseudo-Riemannian geometry, together with a few elements of mathematical relativity. in addition to his own learn articles, he was once excited by numerous textbooks and learn monographs.
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Additional resources for An Introduction to Differential Manifolds
For the (more diﬃcult) proof in the case where m > n, see for example [Hirsch 76] or [Golubitsky-Guillemin 73]. 9. Diﬀerential Calculus in Inﬁnite Dimensions The notion of diﬀerential and the chain rule extends word for word to the case of normed vector spaces on the condition that we require the diﬀerential L to be a continuous linear map. The inverse function theorem extends as well to Banach spaces (we must of course suppose that the diﬀerential has a continuous inverse3 ). Indeed the proof rests essentially on the ﬁxed point theorem for contraction mappings, valid for all complete metric spaces.
3. 22 a) A parametrization of a p-dimensional submanifold M of Rn is a map from an open subset Ω in Rp to Rn that is simultaneously an immersion in Rn and a homeomorphism of Ω to an open subset of M . b) A local parametrization is a map from Ω to Rn that induces a parametrization in a neighborhood of every point of Ω. 21 every submanifold can be covered by open subsets that are images of parametrizations. Examples a) The map t → (cos t, sin t) from R to R2 is a local parametrization of the circle x2 + y 2 = 1.
Y m ) = y 1 , . . , y r , y r+1 + k 1 (y 1 , . . , y r ), y m + k m−r (y 1 , . . , y r ) . Show that h deﬁnes a diﬀeomorphism between two open neighborhoods of 0 in Rm , and that the map f2 deﬁned in a neighborhood of 0 of Rm by h(f2 (x)) = f1 (x) is of the form f2 (x1 , . . , xn ) = (x1 , . . , xr , 0, . . , 0). Deduce for all y0 ∈ f (Rn ), the preimage f −1 (y0 ) is a submanifold of dimension n − r in Rn . Application. By considering the map f deﬁned by f (M ) = tM M , show that O(n) = M ∈ GL(n, R) : tM = M −1 is a submanifold of dimension n(n−1) 2 in GL(n, R).
An Introduction to Differential Manifolds by Jacques Lafontaine