Abstract Nonsense

Crushing one theorem at a time

Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms

Point of Post: In this post we discuss an important consequence of the inverse function theorem which relates local diffeomorphisms to global diffeomorphisms.

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Local to Global Diffeomorphism

Let U\subseteq\mathbb{R}^n be open and f:U\to\mathbb{R}^m. We call f a local diffeomorphism if for every x\in U there exists a neighborhood V of x such that f(V) is open in \mathbb{R}^m and the map f_{\mid V}:V\to f(V) is a diffeomorphism. We define local homeomorphisms similarly. We note the obvious theorem
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Theorem: Let U\subseteq\mathbb{R}^n be open. Then, f:U\to\mathbb{R}^m is a homeo(diffeo)morphism if and only if it’s a bijective local homeo(diffeo)morphism.

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Which is true by definition since differentiability is a local property. The important fact we’d like to note is the following:

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Theorem: Let f:U\to\mathbb{R}^m be a local homeomorphism, then f is an open map.

Proof: Let V\subseteq U be open. We know that for each x\in V there is neighborhood O_x\subseteq U for which f(O_x) is open in \mathbb{R}^m and f_{\mid O_x}:O_x\to f(O_x) is a homeomorphism. Note then that

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\displaystyle f(V)=f\left(\bigcup_{x\in V}\left(O_x\cap V\right)\right)=\bigcup_{x\in V}f(O_x\cap V)

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But, since O_x\cap V is open in O_x and f is a homeomorphism we have that f(O_x\cap V) is open in O_x, but since O_x is open in \mathbb{R}^m this implies that f(O_x\cap V) is also open in \mathbb{R}^m and so f(V) is the union of open sets in \mathbb{R}^m and so trivially open. \blacksquare

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So, what does this have to do with the inverse function theorem? Well, the following:

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Theorem: Let f:U\to\mathbb{R}^n, with U\subseteq\mathbb{R}^n open, be such that the total derivative D_f(x_0) is invertible for each x_0\in U. Then, f is a local diffeomorphism and so, in particular, open.

Proof: This is an immediate consequence of the inverse function theorem which says that since D_f(x_0) is invertible x_0 has a neighborhood for which the restriction of f to this neighborhood is a diffeomorphism. \blacksquare

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1.  Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

2. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.


September 22, 2011 - Posted by | Analysis, Topology | , , , , , , , ,


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