By Pedro M. Gadea, Jaime Muñoz Masqué, Ihor V. Mykytyuk

ISBN-10: 9400759517

ISBN-13: 9789400759510

ISBN-10: 9400759525

ISBN-13: 9789400759527

This is the second one variation of this top promoting challenge publication for college kids, now containing over four hundred thoroughly solved routines on differentiable manifolds, Lie conception, fibre bundles and Riemannian manifolds.

The workouts move from straight forward computations to fairly subtle instruments. some of the definitions and theorems used all through are defined within the first part of every one bankruptcy the place they appear.

A 56-page selection of formulae is incorporated which are necessary as an aide-mémoire, even for academics and researchers on these topics.

In this second edition:
• seventy six new difficulties
• a bit dedicated to a generalization of Gauss’ Lemma
• a brief novel part facing a few houses of the strength of Hopf vector fields
• an elevated selection of formulae and tables
• a longer bibliography

Audience

This e-book might be priceless to complex undergraduate and graduate scholars of arithmetic, theoretical physics and a few branches of engineering with a rudimentary wisdom of linear and multilinear algebra.

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Extra info for Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers

Example text

40 Let S be the subset of R2 which consists of all the points of the set U = {(s, 0)}, s ∈ R, and the point (0, 1). Let U1 be the set obtained from U replacing the point (0, 0) by the point (0, 1). We define the maps ϕ : U → R, ϕ(s, 0) = s, ϕ1 : U1 → R, ϕ1 (s, 0) = s, s = 0, ϕ1 (0, 1) = 0. Prove that {(U, ϕ), (U1 , ϕ1 )} is a C ∞ atlas on S, but S is not Hausdorff with the induced topology. 26 1 Differentiable Manifolds Fig. 14 The straight line with a double point Solution U ∪ V = S, ϕ and ϕ1 are injective maps in R, and the changes of coordinates ϕ ◦ ϕ1−1 and ϕ1 ◦ ϕ −1 are both the identity on the open subset R \ {0}.

X k = x i ei , where {ei } is a basis of L. By virtue of Sard’s Theorem, f (Rk ) = L has zero measure. 56 Let M1 and M2 be two C ∞ manifolds. Give an example of differentiable mapping f : M1 → M2 such that all the points of M1 are critical points and the set of critical values has zero measure. Solution Let f : M1 → M2 defined by f (p) = q, for every p ∈ M1 and q a fixed point of M2 . Then the rank of f is zero, hence all the points of M1 are critical. On the other hand, the set of critical values reduces to the point q, and the set {q} has obviously zero measure.

In fact, ϕ −1 (s) = f ϕ −1 (s) = (0, 1 − s) (sin 2πs, cos 2πs) if − 1 < s < 0, if 0 s < 1, (0, 1 − s) (sin 2π(1 − s), cos 2π(1 − s)) if − 1 < s < 0, if 0 s < 1, and ψ ◦ f ◦ ϕ −1 (s) = s, s ∈ (−1, 1). 67 The aim of the present problem is to prove that the manifold of affine straight lines of the plane, the 2-dimensional real projective space minus a point, and the infinite Möbius strip are diffeomorphic. Explicitly: (a) Let M denote the set of affine straight lines of the plane, that is, M = r(a, v) : a, v ∈ R2 , v = 0 , where r(a, v) = {a + tv : t ∈ R} ⊂ R2 is the (affine) straight line of R2 determined by a and v.

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Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers by Pedro M. Gadea, Jaime Muñoz Masqué, Ihor V. Mykytyuk


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