By Allan J. Sieradski

This article is an creation to topology and homotopy. themes are built-in right into a coherent entire and built slowly so scholars aren't crushed. the 1st half the textual content treats the topology of whole metric areas, together with their hyperspaces of sequentially compact subspaces. the second one 1/2 the textual content develops the homotopy type. there are various examples and over 900 routines, representing quite a lot of hassle. This publication will be of curiosity to undergraduates and researchers in arithmetic.

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Pull-back of bundles 41 local map f defined around x with values in G. We denote f (x) by a and let A ∈ Ta G be the image by f∗ of some X ∈ Tx M . The Leibniz rule yields σ∗ (X) = (Ra )∗ (σ∗ (X)) + (Lu )∗ (A). By assumption σ∗ (X) and σ∗ (X) both belong to H, and since H is Ginvariant, we obtain (Lu )∗ (A) ∈ H. On the other hand (Lu )∗ (A) ∈ V by definition, so (Lu )∗ (A) = 0. Since Lu is a diffeomorphism, we must have A = 0. The expression of the section ψ in the frame σ is ψ = [σ , ρ(f −1 )ξ], so using the Leibniz rule again [σ (x), ∂X (ρ(f −1 )ξ)] = [σ (x), ρ(f −1 )(∂X ξ)] − [σ (x), ρ∗ (A)ξ] = [σ(x), ∂X ξ].

Vk ) of Ex (the image by u of the canonical basis of Rk ). For every x ∈ M and tangent vector X ∈ Tx M , we can define the horizontal lift of X to any u = (v1 , . . , vk ) ∈ Glx (E) in the following way. Take local sections σi of E around x such that σi (x) = vi and (∇X σi )x = 0. Then σ := (σ1 , . . , σk ) is a local section of Gl(E) satisfying σ(x) = u. We define ˜ u := σ∗ (X) ∈ Tu Gl(E). It can be easily shown, using a local trivialization (X) ˜ does not depend on the local sections σi .

8. A vector field on a Riemannian manifold is called a Killing vector field or infinitesimal isometry if its local flow consists of (local) isometries of M . 9. Let ξ be a vector field on a Riemannian manifold (M, g, ∇). The following statements are equivalent: (i) ξ is Killing with respect to g. (ii) The Lie derivative of g with respect to ξ vanishes: Lξ g = 0. (iii) The covariant derivative ∇ξ is skew-symmetric with respect to g: g(∇X ξ, Y ) + g(∇Y ξ, X) = 0, ∀ X, Y ∈ T M. 4) Proof. Let ϕt denote the local flow of ξ.