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Unformatted text preview: 2.2. TANGENT BUNDLE 35 • Thus, every fiber bundle is an associated bundle with some principal bun dle. So, all bundles can be constructed as associated bundles from principal bundles. All we need is the structure group. The fiber is not important. • A vector bundle is a fiber bundle whose fiber is a vector space. • A section of a bundle ( E , π, M ) is a map s : M → E such that the image of each point p ∈ M is in the fiber π 1 ( p ) over this point, that is, s ( p ) ∈ π 1 ( p ), or π ◦ s = Id M . 2.2.2 Tangent Bundle • Definition 2.2.1 Let M be a smooth manifold. The tangent bundle T M to M is the collection of all tangent vectors at all points of M. T M = { ( p , v )  p ∈ M , v ∈ T p M } • Let dim M = n . • Let p ∈ M be a point in the manifold M , ( U , x ) be a local chart and ( x i ) be the local coordinates of the point p . • Let ∂ i = ∂/∂ x i be the coordinate basis for T p M ....
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 Spring '10
 Wong
 Geometry, Vector Space, Manifold, Fiber bundle, Differential topology, Tangent bundle

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