Splitting lemma (functions) explained

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:( R ^{n,} 0 ) \to ( R , 0 ) be a smooth function germ, with a critical point at 0 (so (\partial f/\partial x _{i)(0)} = 0 for i = 1 , ... , n ). Let V be a subspace of 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 thatf\circ\Phi(x,y) = (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

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