The Unapologetic Mathematician

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.

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