differential geometry w notes from teacher_Part_17

differential geometry w notes from teacher_Part_17 - 2.2....

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Unformatted text preview: 2.2. TANGENT BUNDLE 33 2.2 Tangent Bundle 2.2.1 Fiber Bundles • Let M and E be smooth manifolds and π : E → M be a smooth map. Then the triple ( E , π, M ) is called a bundle . • The manifold M is called the base manifold of the bundle and the manifold E is called the bundle space of the bundle (or the bundle space manifold ). The map π is called the projection . • The inverse image π- 1 ( p ) of a point p ∈ M is called the fiber over p . • The projection map is supposed to be surjective, that is, the di ff erential π * has the maximal rank equal to dim M . • Let { U α } α ∈ A be an atlas of local charts covering the base manifold M and let U αβ = U α ∩ U β , U αβγ = U α ∩ U β ∩ U γ etc. • A fiber bundle is a bundle all fibers of which, π- 1 ( p ), ∀ p ∈ M , are di ff eo- morphic to a common manifold F called the typical fiber of the bundle (or just the fiber )....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_17 - 2.2....

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