Let be a smooth manifold, a real valued function.
is a critical point of if for any curve with , the vector
The critical point is regular if for one (or equivalently any) chart , where and , the Hessian matrix
is a nondegenerate (i.e. maximal rank) matrix.
Morse lemma states that for any regular critical point of there is a chart around such that the function in these coordinates is quadratic:
and number is determined by the Hessian matrix. While the Morse lemma is proved by Morse, the modern proof is by the Moser’s deformation method. The Morse lemma can be generalized to smooth functions on a Hilbert manifold, in which case there is a linear operator such that in suitable local coordinates, quadratic functional can be written as <>.