By P M Gadea; J Muñoz Masqué; I V Mikiti︠u︡k

ISBN-10: 9400759525

ISBN-13: 9789400759527

Differentiable Manifolds -- Tensor Fields and Differential types -- Integration on Manifolds -- Lie teams -- Fibre Bundles -- Riemannian Geometry -- a few formulation and Tables

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**Sample text**

Let f : S 2 → S 2 be the map induced by the automorphism of R3 with matrix √ ⎞ ⎛√ 2/2 0 2/2 ⎝ 0 ⎠. 1 √0 √ 2/2 − 2/2 0 Consider the coordinate neighbourhood U = (x, y, z) ∈ S 2 : x + z = 0 . ∂ ∂ Compute f∗ ( ∂θ |p ) and f∗ ( ∂ϕ |p ) for p ≡ (θ0 , ϕ0 ) ∈ U such that f (p) also belongs to U . 32 1 Differentiable Manifolds Solution This parametrisation can be described by saying that we have a chart Φ from U to an open subset of A = (0, π) × (0, 2π) with Φ −1 (u, v) = (sin u cos v, sin u sin v, cos u), u, v ∈ A, and that we call θ = u ◦ Φ, ϕ = v ◦ Φ, where u and v are the coordinate functions on A.

12 Two charts which do not define an atlas identity map on ψ(U ∩ V ) = (−∞, 0) = ϕ(U ∩ V ), and ϕ(U ∩ V ), ψ(U ∩ V ) are open subsets of R. Hence A = {(U, ϕ), (V , ψ)} is a C ∞ atlas on M. (ii) The map γ is injective, and γ (V ) = R \ {0}, γ (U ∩ V ) = (−∞, 0), are open subsets of R. Moreover, the maps γ ◦ ϕ −1 , ϕ ◦ γ −1 , γ ◦ ψ −1 , and ψ ◦ γ −1 are C ∞ maps. Thus, γ is, in fact, a chart of the above differentiable structure. 38 Let S = (x, 0) ∈ R2 : x ∈ (−1, +1) ∪ (x, x) ∈ R2 : x ∈ (0, 1) . Let U = (x, 0) : x ∈ (−1, +1) , ϕ : U → R, ϕ(x, 0) = x, V = (x, 0) : x ∈ (−1, 0] ∪ (x, x), x ∈ (0, 1) , ψ : V → R, ψ(x, 0) = x, ψ(x, x) = x (see Fig.

36 1 Differentiable Manifolds Fig. 18 The graph of the map t → (t 2 , t 3 ) Solution Let dim L = k n − 1. Consider the map f x 1 , x 2 , . . , x k = x i ei , f : R k → Rn , 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 .

### Analysis and algebra on differentiable manifolds : a workbook for students and teachers by P M Gadea; J Muñoz Masqué; I V Mikiti︠u︡k

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