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.

\text{ }

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
\text{ }

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.

\text{ }

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

\text{ }

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

\text{ }

\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)

\text{ }

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

\text{ }

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

\text{ }

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

\text{ }

\text{ }

References:

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.

Advertisements

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

2 Comments »

  1. […] the property that for all . We know then is injective and a local diffeomorphism, and so from analysis we know that is an open map and so for some open interval . So, consider the reparamaterization […]

    Pingback by Reparamaterization, Regular Curves, and Unit Speed Curves (Pt. II) « Abstract Nonsense | September 23, 2011 | Reply

  2. […] is the plane. Note though, that since is open it contains a neighborhood of in and since we know diffeomorphisms are open (since is open) we can conclude that is an interior point in when […]

    Pingback by Instructive Non-Examples « Abstract Nonsense | October 14, 2011 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: