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N so that (t ◦ si )(p) = (p, hi (p)) for p ∈ U. , hn (p) as column vectors for p ∈ U. Combining this, we have that n n i=1 i=1 t ◦ f −1 (p, a) = t ( ∑ ai si (p)) = (p , ∑ ai hi (p)) = (p, h(p) a) and hence also f ◦ t −1 (p, b) = (p, h(p)−1 b) for every p ∈ U and a, b ∈ Rn . Since this holds for every local trivialization (t, π −1 (U)) in E, it follows that f is an equivalence over M. , sn (p) ∈ π −1 (p) are linearly independent for every p ∈ U. , n, where ei is the i-th standard basis vector in Rn .

52. 54 can be generalized in the following way. Let π : E → M be an n-dimensional vector bundle, and let f : N → M be a smooth map from a smooth manifold N . Then a smooth map F : N → E with π ◦ F = f is called a section of π along f , or a lifting of f to E. The set of all liftings of f to E is denoted by Γ( f ; E) . 45 that pr1 ◦ t ◦ F = fU . Hence F| f −1 (U) is completely determined by the map a : f −1 (U) → Rn , called the local representation of F on U , given by (t ◦ F)(p) = ( f (p), a(p)) f −1 (U) .

40 Corollary Let A be a subset of a smooth manifold N, and fix a topology on A. Then there is at most one smooth structure on A which makes it an immersed submanifold of N. PROOF : Let P and M be equal as topological spaces and let each have a smooth structure which makes it an immersed submanifold of N. 37 a diffeomorphism proving that P and M must have the same smooth structure. 41 Proposition Let f : M n → N m be smooth, and let q be a point in N. If f has constant rank k in a neighbourhood of f −1 (q), then f −1 (q) is a closed submanifold of M of dimension n − k.

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