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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 one-to-one, 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 n-dimensional 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 : U X V . Intuitively, the set X is an n-dimensional 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 = . Theorem 6.9. The set X = f 1 (0) is an n-dimensional 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 = . (6.7) Recall that R N = R R n and : R N R k is the map that sends...
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- Fall '04