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Splitting lemma (functions)

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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.

Contents

Formal statement

Let f : ( R n , 0 ) → ( R , 0 ) be a smooth function germ, with a critical point at 0 (so ( ∂ f / ∂ x i ) ( 0 ) = 0 , ( 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 Φ ( x , y ) of the form Φ ( x , y ) = ( ϕ ( x , y ) , y ) with x ∈ V , y ∈ W , and a smooth function h on W such that

f ∘ Φ ( x , y ) = 1 2 x T B x + 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

Splitting lemma (functions) Wikipedia
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