An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This booklet is an creation to differential manifolds. It offers reliable preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie thought. It presupposes little historical past: the reader is just anticipated to grasp simple differential calculus, and a bit point-set topology. The booklet covers the most themes of differential geometry: manifolds, tangent area, vector fields, differential kinds, Lie teams, and some extra subtle themes resembling de Rham cohomology, measure conception and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide stable foundations. particularly, the creation of “abstract” notions akin to manifolds or differential varieties is prompted 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. strategies are supplied for many of them.

The booklet may 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 classification 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 shortly emeritus. His major study pursuits are Riemannian and pseudo-Riemannian geometry, together with a few facets of mathematical relativity. in addition to his own examine articles, he was once serious about a number of textbooks and examine monographs.

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Show that R2 in R2 . (Euler’s identity). {0} is diffeomorphic to the complement of a closed ball 6. Cusps of the “second kind” and diffeomorphisms Show that the map f : (x, y) → (x, y − x2 ) is a local diffeomorphism in a neighborhood of 0. Sketch the curve t → (t2 , t4 + t5 ) and its transformation under f . What can you say? 7. , a curve without double points) under the map z → z 2 of C to C, a) not surrounding the origin; b) surrounding the origin; c) passing through the origin. 8*. Cartan’s decomposition of the linear group Consider Rn equipped with an inner product.

2. Laplacian and isometries Suppose f is a C 2 function from an open subset of Rn to R. The Laplacian of f , denoted Δf , is defined by n ∂i2 f. Δf = i=1 a) Suppose that f is defined on all of Rn (or on Rn {0}), and depends only on the distance to the origin, in other words there exists a function φ defined on R+ (or R+ − 0) such that f (x) = φ( x ). Show that φ is of class C 2 and calculate the derivatives in terms of φ. b*) Characterize the linear maps A from Rn to Rn such that for all C ∞ functions f , we have Δf ◦ A = Δ(f ◦ A).

To see that i) implies ii), we consider the components (f i )1 i n of f . By hypothesis, their differentials are linearly independent at every point of U . Set g = (f p+1 , . . , f n ). We then have a submersion of U to Rn−p such that M ∩ U = g −1 (0). Now suppose that iii) is true. 17, we may replace Ω by a smaller open subset and find a diffeomorphism ϕ from an open subset U containing h(0) = a to Rn , such that (ϕ ◦ h)(x1 , . . , xp ) = (x1 , . . , xp , 0, . . , 0). Then ϕ(U ∩ M ) = ϕ(h(Ω)) = ϕ(U ) ∩ (Rp × {0}).

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