Splitting lemma (functions)

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See also 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 be a smooth function germ, with a critical point at 0 (so ). Let V be a subspace of 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 of the form with , and a smooth function h on W such that

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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