The Unapologetic Mathematician

Mathematics for the interested outsider

The Inverse Function Theorem

Recall the inverse function theorem from multivariable calculus: if f:U\to\mathbb{R}^n is a C^1 map defined on an open region U\subseteq\mathbb{R}^n, and if the Jacobian of f has maximal rank n at a point p\in U then there is some neighborhood V of p so that the restriction f\vert_V:V\to f(V)\subseteq\mathbb{R}^n is a diffeomorphism. This is slightly different than how we stated it before, but it’s a pretty straightforward translation.

Anyway, this generalizes immediately to more general manifolds. We know that the proper generalization of the Jacobian is the derivative of a smooth map f:U\to N, where U\subseteq M is an open region of an n-manifold and N is another n-manifold. If the derivative f_{*p}:\mathcal{T}_pM\to\mathcal{T}_{f(p)}N has maximal rank n at p, then there is some neighborhood V\subseteq M of p for which f\vert_V:V\to f(V)\subseteq N is a diffeomorphism.

Well, this is actually pretty simple to prove. Just take coordinates x at p\in M and y at f(p)\in N. We can restrict the domain of f to assume that U is entirely contained in the x coordinate patch. Then we can set up the function y\circ f\circ x^{-1}:x(U)\to\mathbb{R}^n.

Since f has maximal rank, so does the matrix of f with respect to the bases of coordinate vectors \frac{\partial}{\partial x^i} and \frac{\partial}{\partial y^j}, which is exactly the Jacobian of y\circ f\circ x^{-1}. Thus the original inverse function theorem applies to show that there is some W\subseteq x(U) on which y\circ f\circ x^{-1} is a diffeomorphism. Since the coordinate maps x and y are diffeomorphisms we can write W=x(V) for some V\subseteq M, and conclude that f:V\to f(V) is a diffeomorphism, as asserted.

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April 14, 2011 - Posted by | Differential Topology, Topology


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