Smooth Manifolds - Main Theorem

The inverse function theorem extends from calculus to manifolds, providing the fundamental tool for understanding local structure. It guarantees that smooth maps with invertible differential are locally diffeomorphisms.

TheoremInverse Function Theorem for Manifolds

Let $f: M \to N$ be a smooth map between $n$ -dimensional manifolds, and let $p \in M$ . If the differential $df_p: T_pM \to T_{f(p)}N$ is an isomorphism, then there exist open neighborhoods $U$ of $p$ and $V$ of $f(p)$ such that $f|_U: U \to V$ is a diffeomorphism.

This theorem is the cornerstone of local differential geometry. It tells us that the linear approximation of a map (its differential) determines its local invertibility, just as in ordinary calculus.

Remark

The theorem requires the differential to be an isomorphism, which automatically forces $\dim M = \dim N$ . For maps between manifolds of different dimensions, we have related results: the implicit function theorem and the submersion/immersion theorems.

TheoremImplicit Function Theorem

Let $f: M \to N$ be a smooth map, and suppose $q \in N$ is a regular value, meaning that for all $p \in f^{-1}(q)$ , the differential $df_p: T_pM \to T_qN$ is surjective. Then $f^{-1}(q)$ is a smooth submanifold of $M$ of dimension $\dim M - \dim N$ .

This powerful result shows that level sets of smooth maps are manifolds under appropriate conditions. It's the key to recognizing manifolds "in the wild" - for instance, spheres as level sets of the function $f(x) = \|x\|^2$ .

ExampleSphere as Level Set

Consider $f: \mathbb{R}^{n+1} \to \mathbb{R}$ given by $f(x) = \|x\|^2 = \sum_{i=1}^{n+1} (x^i)^2$ . The differential at any $x \neq 0$ is $df_x(v) = 2\langle x, v \rangle$ , which is surjective. Thus $S^n = f^{-1}(1)$ is a smooth $n$ -manifold.

DefinitionSubmersion and Immersion

A smooth map $f: M \to N$ is a submersion at $p$ if $df_p$ is surjective, and an immersion at $p$ if $df_p$ is injective. The map is a submersion (immersion) if it is so at every point.

TheoremConstant Rank Theorem

Let $f: M \to N$ be a smooth map. If $df_p$ has constant rank $k$ for all $p$ in a neighborhood of $p_0$ , then there exist charts $(U, \varphi)$ at $p_0$ and $(V, \psi)$ at $f(p_0)$ such that the coordinate representation of $f$ is

\psi \circ f \circ \varphi^{-1}(x^1, \ldots, x^m) = (x^1, \ldots, x^k, 0, \ldots, 0)

This remarkable theorem says that locally, any smooth map of constant rank looks like a projection in suitable coordinates - a fundamental simplification that underlies many geometric constructions.

Remark

The constant rank theorem encompasses both the inverse function theorem (when $k = m = n$ ) and the implicit function theorem (when the rank equals $\dim N$ ). It provides a unified framework for understanding the local structure of smooth maps.