A smooth real valued function on a smooth manifold is called a Morse function if every critical point of is regular.
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.
A choice of a Morse function on a compact manifold is often used to study topology of the manifold. This is called the Morse theory.
One of the basic tools is Morse lemma.
See also perfect Morse function and wikipedia: Morse theory
Revised on April 6, 2010 23:04:37
by Zoran Škoda