lecture31

lecture31 - Lecture 31 6.3 Examples of Manifolds We begin...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 4

lecture31 - Lecture 31 6.3 Examples of Manifolds We begin...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online