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Unformatted text preview: Math 598 Jan 21, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 4 2 Differentiable manifolds 2.1 Differential structures and maps Recall that a mapping f : R n → R m is differentiable of class C r provided that all of its partial derivatives exist and are continuous up to and including order r . If all partial derivatives of f exist up to any order we say that f is C ∞ or smooth . A collection { ( U i , φ i ) } i ∈ I is called an atlas of a manifold M , if U i cover M and φ : U i → R n are homeomorphisms. We say that this atlas is C r if for every i , j ∈ I , φ i ◦ φ − 1 j : φ j ( U i ∩ U j ) → φ i ( U i ∩ U j ) is C r . We say that M is a C r manifold if it admits a C r atlas. Exercise 1. Show that S n is a smooth ( C ∞ ) manifold. Exercise 2. Show that RP n is a smooth manifold ( Hint : Define RP n as the quotient space ( R n +1 − { o } ) / ∼ , where x ∼ y iff x = λy for some λ 6 = 0....
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 Spring '08
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 Geometry, Topology, Derivative, Manifold, Topological space, smooth manifold

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