An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This booklet is an creation to differential manifolds. It offers stable preliminaries for extra complicated issues: Riemannian manifolds, differential topology, Lie idea. It presupposes little heritage: the reader is barely anticipated to grasp easy differential calculus, and a bit point-set topology. The ebook covers the most issues of differential geometry: manifolds, tangent area, vector fields, differential varieties, Lie teams, and some extra refined subject matters reminiscent of de Rham cohomology, measure thought and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide strong foundations. specifically, the advent of “abstract” notions similar to manifolds or differential varieties is encouraged through questions and examples from arithmetic or theoretical physics. greater than a hundred and fifty workouts, a few of them effortless and classical, a few others extra refined, might help the newbie in addition to the extra specialist reader. strategies are supplied for many of them.

The publication could be of curiosity to numerous 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 creation aux variétés différentielles has been a best-seller in its type in France for lots of years.

Jacques Lafontaine used to be successively assistant Professor at Paris Diderot college 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. along with his own study articles, he was once fascinated with numerous textbooks and examine monographs.

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Extra resources for An Introduction to Differential Manifolds

Example text

Dxn = D gΔf dx1. . dxn . D See Exercise 14 in Chapter 6, where one also shows, with the same hypothesis on f that |dfx |2 dx1. . dxn . f Δf dx1. . dxn = D D 2. “Formally” because the operator is not defined on the whole space. 36, and its proof rests, modulo adequate functional analysis, on a study of the extrema of the functional D |dfx |2 dx1. . dxn on the unit sphere of L2 (D). See for example [Attouch-Buttazzo-Michaille 06, Chapter 7]. This variational method also gives a way to control the volume of parallelepipeds in Euclidean space.

We use the identity k k Ai B k−i (A + B)k = i i=0 which is true if A and B commute. iii) is immediate from i) after passing to the limit, as for every integer k, P −1 Ak P = (P −1 AP )k . iv) The property is evident for diagonal matrices, and by iii) for diagonalizable matrices (and even for real matrices that are C-diagonalizable), which form a dense subset of End(K n )). Therefore the property holds for all matrices by continuity of exp and det (see Exercise 24 for another proof). v) is clear.

Suppose (v1 , . . , vn ) is an n-tuple of vectors for which the continuous map (a1 , . . , an ) −→ | det(a1 , . . , an )| attains it maximum. In a neighborhood of this n-tuple, this map is equal to the determinant (or its negative). 35 applied to the partial functions ai −→ det(a1 , . . , an ), for all i, the determinant det(v1 , . . , vi−1 , h, vi+1 , . . , (vi )⊥ ⊂ span(v1 , . . , vi−1 , vi+1 , . . , vn ). 36 An Introduction to Differential Manifolds Since these two subspaces are of codimension 1, they are therefore equal, and vi is orthogonal to each vk for k = i.

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