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Unformatted text preview: Lecture 31 6.3 Examples of Manifolds We begin with a review of the definition of a manifold. Let X be a subset of R n , let Y be a subset of R m , and let f : X Y be a → continuous map. Definition 6.6. The map f is C ∞ if for every p ∈ X , there exists a neighborhood U p of p in R n and a C ∞ map g p : U p → R m such that g p = f on U p ∩ X . Claim. If f : X Y is continuous, then there exists a neighborhood U of X in R n → and a C ∞ map g : U → R m such that g = f on U ∩ X . Definition 6.7. The map f : X → Y is a diffeomorphism if it is onetoone, onto, and both f and f − 1 are C ∞ maps. We define the notion of a manifold. Definition 6.8. A subset X of R N is an ndimensional manifold if for every p ∈ X , there exists a neighborhood V of p in R N , an open set U in R n , and a diffeomorphism φ : U → X ∩ V . Intuitively, the set X is an ndimensional manifold if locally near every point p ∈ X , the set X “looks like an open subset of R n .” Manifolds come up in practical applications as follows: Let U be an open subset of R N , let k < N , and let f : R N → R k be a C ∞ map. Suppose that 0 is a regular value of f , that is, f − 1 (0) ∩ C f = φ . Theorem 6.9. The set X = f − 1 (0) is an ndimensional manifold, where n = N − k . Proof. If p ∈ f − 1 (0), then p / ∈ C f . So the map Df ( p ) : R N R k is onto. The map → f is a submersion at p . By the canonical submersion theorem, there exists a neighborhood V of 0 in R n , a neighborhood U 0 of p in U , and a diffeomorphism g : V U such that → f ◦ g = π. (6.7) Recall that R N = R × R n and π : R N R k is the map that sends...
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 Fall '04
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 Topology, Open set

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