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

f\circ\Phi(x,y) = \textstyle\frac12 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

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 .

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