differential geometry w notes from teacher_Part_18

# differential geometry w notes from teacher_Part_18 - 2.2...

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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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differential geometry w notes from teacher_Part_18 - 2.2...

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