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.

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

Formal statement

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

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

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.