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 formula_1 be a smooth function germ, with a critical point at 0 (so formula_2 for formula_3). Let "V" be a subspace of formula_4 such that the restriction "f" | 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 formula_5 of the form formula_6 with formula_7, 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, ...