differential geometry w notes from teacher_Part_9

differential geometry w notes from teacher_Part_9 - 1.3...

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Unformatted text preview: 1.3. TANGENT VECTORS AND MAPPINGS 17 1.3.4 Mappings • Let M be an n-dimensional manifold and N be an m-dimensional manifold and let F : M → N be a map from M to N . Let ( U α , ϕ α ) α ∈ A be an atlas in M and ( V β , ψ β ) β ∈ B be an atlas in N . We define the maps F αβ = ψ β ◦ F ◦ ϕ- 1 α : ϕ α ( U α ) → ψ β ( V β ) from open sets in R n to R m defined by y a β = F a αβ ( x 1 α , . . . , x n α ) , where a = 1 , . . . , m . • The map F is said to be smooth if F a αβ are smooth functions of local coor- dinates x i α , i = 1 , . . . , n . • The process of replacing the map F by the functions F αβ = ψ β ◦ F ◦ ϕ- 1 α is usually omitted, and the maps F and F αβ are identified, so that we think of the map F directly in terms of local coordinates. • If n = m and the map F : M → N is bijective and both F and F- 1 are di ff erentiable, then F is called a di ff eomorphism ....
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differential geometry w notes from teacher_Part_9 - 1.3...

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