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 [math]\displaystyle{ f:(\mathbb{R}^n, 0) \to (\mathbb{R}, 0) }[/math] be a smooth function germ, with a critical point at 0 (so [math]\displaystyle{ (\partial f/\partial x_i)(0) = 0 }[/math] for [math]\displaystyle{ i = 1, \dots, n }[/math]). Let V be a subspace of [math]\displaystyle{ \mathbb{R}^n }[/math] 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 [math]\displaystyle{ \Phi(x, y) }[/math] of the form [math]\displaystyle{ \Phi(x, y) = (\phi(x, y), y) }[/math] with [math]\displaystyle{ x \in V, y \in W }[/math], and a smooth function h on W such that

[math]\displaystyle{ f\circ\Phi(x,y) = \frac{1}{2} x^TBx + h(y). }[/math]

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

• Poston, Tim; Stewart, Ian (1979), Catastrophe Theory and Its Applications, Pitman, ISBN 978-0-273-08429-7 .
• Brocker, Th (1975), Differentiable Germs and Catastrophes, Cambridge University Press, ISBN 978-0-521-20681-5 .