Splitting lemma (functions)

Formal statement

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