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 |_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, ...

References

{{citation|first1=Tim|last1=Poston|first2=Ian|last2=Stewart|authorlink2=Ian Stewart (mathematician)|title=Catastrophe Theory and Its Applications|publisher=Pitman|year=1979|ISBN=978-0-273-08429-7}}.
{{citation|first=Th|last=Brocker|title=Differentiable Germs and Catastrophes|publisher=Cambridge University Press|year=1975|ISBN=978-0-521-20681-5}}.

Category:Singularity theory

Category:Functions and mappings