Splitting lemma (functions)

Not to be confused with the splitting lemma in homological algebra.

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

$f\circ \Phi (x,y)={\frac {1}{2}}x^{T}Bx+h(y).$

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

• 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.