Splitting lemma (functions)

In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point.

Formal statement
Let $$f:(\mathbb{R}^n, 0) \to (\mathbb{R}, 0)$$ be a smooth function germ, with a critical point at 0 (so $$(\partial f/\partial x_i)(0) = 0$$ for $$i = 1, \dots, n$$). Let V be a subspace of $$\mathbb{R}^n$$ such that the restriction f&thinsp;|V is non-degenerate, and write B for the Hessian matrix of this restriction. Let W be any complementary subspace to V. Then there is a change of coordinates $$\Phi(x, y)$$ of the form $$\Phi(x, y) = (\phi(x, y), y)$$ with $$x \in V, y \in W$$, and a smooth function h on W such that
 * $$f\circ\Phi(x,y) = \frac{1}{2} x^TBx + h(y).$$

This result is often referred to as the parametrized Morse lemma, which can be seen by viewing y as the parameter. It is the gradient version of the implicit function theorem.

Extensions
There are extensions to infinite dimensions, to complex analytic functions, to functions invariant under the action of a compact group, ...