An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This e-book is an advent to differential manifolds. It offers reliable preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie concept. It presupposes little historical past: the reader is simply 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 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 provide reliable foundations. specifically, the creation of “abstract” notions comparable to manifolds or differential kinds is stimulated 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 refined, may also help the newbie in addition to the extra professional reader. ideas are supplied for many of them.

The booklet might 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 classification in France for plenty 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 shortly emeritus. His major learn pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. in addition to his own learn articles, he used to be fascinated with a number of textbooks and learn monographs.

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Notice that, in addition to being closed under sums and real scalar multiples, R4 is actually closed under matrix multiplication since α β −β¯ α ¯ γ δ −δ¯ γ¯ = αγ − β δ¯ αδ + β¯ γ −(αδ + β¯ γ) αγ − β δ¯ . 9) which is in R4 . Thus, under matrix multiplication, the nonzero elements of R4 take on the structure of a (non-Abelian) group. 10) and that 1 is a multiplicative identity. 11) below) completely determine the multiplication on R4 . 10) and 1x = x1 = x for all x ∈ H. 12) below and apply the natural isomorphism).

The converse is a simple exercise in complex arithmetic: If (z 1 , z 2 ), (w1 , w2 ) ∈ S 3 and z 1 /z 2 = w1 /w2 , then there exists a g ∈ U (1) such that (w1 , w2 ) = (z 1 g, z 2 g) = (z 1 , z 2 ) · g. Thus, the orbits are in one-to-one correspondence with the elements of C∗ and these, via stereographic projection, are in one-to-one correspondence with the elements of S 2 . Let us make this a bit more formal: Define a map P : S 3 → S 2 by z1 −1 P(z 1 , z 2 ) = (ϕ∗S ) . 2) for all (z 1 , z 2 ) ∈ S 3 .

Moreover, if α corresponds to the vector field α, then dα corresponds to the curl of α. “3-forms” on U result from yet one more extension of the wedge product and exterior differentiation operators. In standard coordinates they have the form f (x, y, z)dx ∧ dy ∧ dz and so are in one-to-one correspondence with C ∞ functions f (x, y, z ) on U . In fact, if Ω is a 2-form corresponding to the vector field Ω, then dΩ corresponds to the divergence of Ω. One of the principal virtues of differential forms is this elegant consolidation of the basic operations of vector calculus (div, grad, curl).

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