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 $f:M\to N$ 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? $T_{p}f:T_{p}M\to T_{f(p)}N$. If $f$ is locally invertible with differentiable inverse, then for all $y$ in some neighborhood of $y$ the functoriality of $T$ implies that ${\mathrm{Id}}_{T_y}=T_y(f^{-1}\circ f)=T_{f(y)}{f}^{-1}\circ T_{y}f$ 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 $R^n$

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

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 $R^n$.

Global statement on manifolds

Let $f:M\to N$ 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 $T_{p}f:T_{p}M\to T_{q}N$ is an epimorphism. The implicit function theorem asserts that $Q=f^{-1}(p)$ is a smooth submanifold of $M$ and the tangent bundle $TM{\mid }_N$ globally splits as $TM{\mid }_N\cong TN\oplus R^n$ 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

$$T_{f(x)}N=T_{f(x)}W+(T_{x}f)(T_{x}X).$$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 $Tf$ 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 $R^n$ (mainly lots of examples):

• Frank Jones, Implicit function theorem, pdf