nLab implicit function theorem

Idea

Implicit function theorems give sufficient conditions for the existence of a differentiable inverse of a germ $f_p$ of a differentiable map $f\colon M \to N$ of smooth manifolds at a point $p$. The invertibility is trivially equivalent to the statement that the germ is a local diffeomorphism of some neighborhood of $p$ to some neighborhood of $f(p)$. If it is invertible, then we can consider the tangent map $T_p f\colon T_p M \to T_{f(p)}N$. If $f$ is locally invertible with differentiable inverse, then for all $y$ in some neighborhood of $y$ the functoriality of $T$ implies that $Id_{T_y} = T_y (f^{-1} \circ f) = T_{f(y)} f^{-1} \circ T_y f$ and alike for $f \circ f^{-1}$ at $f(y)$, demonstrating that $T_y f$ must then be invertible. The inverse function theorem says that the invertibility of $T_p f$ is in fact sufficient for the invertibility of the germ, which is then automatically differentiable.

In $\mathbf{R}^n$

Let $U \subset \mathbf{R}^n$ be an open set in a cartesian space, $a\in U$, $f\colon U \to \mathbf{R}^n$ a map of class $C^1$ and $det\left(\frac{\partial f_i}{\partial x_j}(a)\right)\neq 0$. Then there are open sets $V \ni a$, $W \ni f(a)$, $V \subset U$ such that $f|_V\colon V \to W$ is a diffeomorphism and for all $y \in W$ and $(f^{-1})'(y) = (f'[f^{-1}(y)])^{-1}$.

Local statement on manifolds

This is the theorem stated in the Idea section; the differentiable germ is assumed to be of class $C^1$ (continuously differentiable). The statement is local, so one can consider it in charts, hence the proof reduces to the case of $\mathbf{R}^n$.

Global statement on manifolds

Let $f\colon M \to N$ be a smooth map of smooth manifolds. A point $q \in N$ is a regular value of $f$ if for every point $p \in f^{-1}(q)$ the differential $T_p f\colon T_p M \to T_q N$ is an epimorphism. The implicit function theorem asserts that $Q = f^{-1}(p)$ is a smooth submanifold of $M$ and the tangent bundle $T M|_N$ globally splits as $T M|_N \cong T N \oplus \mathbf{R}^n$ where $n = dim N$.

More generally, if $W \subset N$ is a submanifold, we say that the map $f$ is transversal along $W$ if for every point $x\in f^{-1}(W)$ there is an equality

$$T_{f(x)} N = T_{f(x)}W + (T_x f)(T_x X) .$$

In particular, $f$ is transveral along every regular value $p \in N$. The implicit function theorem asserts that the preimage $f^{-1}(W)$ is a smooth submanifold of $M$, the normal bundle $\nu(f^{-1}(W) \subset M)$ is isomorphic to $f^*(\nu(W\subset N))$, and the differential $T f$ exhibits the fiberwise isomorphism $\nu(f^{-1}(W)\subset M)\to \nu(W\subset N)$.

References

• L. H. Loomis, S. Sternberg, Advanced calculus, 1968, 1990 (3.11 in 1990 edition)

• S. Lang, Analysis I

Various applications and related theorems can be found in chapter 5: Local and tangential properties of

• T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; $\exists$ also 1990 German 2nd edition)

An invariant global statement on manifolds is at page 44 of

• А. С. Мищенко, Векторные расслоенния и их применения, Moscow, Nauka 1984

Elementary course notes of the case in $\mathbf{R}^n$ (mainly lots of examples):

• Frank Jones, Implicit function theorem, pdf

On the implicit function theorem in the generality of Fréchet manifolds:

• Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982) 65-222 [doi:1982-07-01/S0273-0979-1982-15004-2]