An Introduction to Differential Manifolds by Jacques Lafontaine PDF

By Jacques Lafontaine

ISBN-10: 3319207350

ISBN-13: 9783319207353

This ebook is an creation to differential manifolds. It supplies strong preliminaries for extra complicated subject matters: Riemannian manifolds, differential topology, Lie concept. It presupposes little historical past: the reader is just anticipated to grasp simple differential calculus, and a bit point-set topology. The publication covers the most issues of differential geometry: manifolds, tangent house, vector fields, differential kinds, Lie teams, and some extra subtle issues akin to de Rham cohomology, measure idea and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide sturdy foundations. specifically, the creation of “abstract” notions reminiscent of manifolds or differential types is prompted 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 subtle, can help the newbie in addition to the extra professional reader. ideas are supplied for many of them.

The booklet will 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 collect a few feeling approximately this gorgeous 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 was once successively assistant Professor at Paris Diderot collage and Professor on the college of Montpellier, the place he's shortly emeritus. His major examine pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. along with his own examine articles, he was once desirous about a number of textbooks and study monographs.

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Remark. Differentiability is defined at a point (more precisely it depends only on the behavior of a function in an arbitrary neighborhood of the point considered). By contrast, the property of being a C p map only makes sense on an open subset. 3. 8. Suppose f is a map from an open subset U of Rm to Rn , and g is a map from an open subset V of Rn to Rp . Suppose that f is differentiable at a ∈ U , with f (a) ∈ V , and further suppose that g is differentiable at f (a). Then g ◦ f is differentiable at a, and d(g ◦ f )a = dgf (a) ◦ dfa .

A C 0 submersion) if after composition on the range side (resp. domain side) with a suitable homeomorphism, it becomes a injective (resp. surjective) linear map. 5. 1. Basic Properties Intuitively, a submanifold of dimension p in Rn is a union of small pieces each of which can each be straightened in a way to form open subsets of Rp . ). 20. A subset M ⊂ Rn is a p-dimensional submanifold of Rn if for all x in M , there exists open neighborhoods U and V of x and 0 in Rn respectively, and a diffeomorphism f : U −→ V such that f (U ∩ M ) = V ∩ (Rp × {0}).

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.

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