Differential Topology and Shape Analysis - Mathematical Tools for Shape Analysis and Description - page 61

C H A P T E R 8

Differential Topology and
Shape Analysis

Morse theory can be seen as the investigation of the relation between functions defined on a man-
ifold and the shape of the manifold itself. e key feature in Morse theory is that information
on the topology of the manifold is derived from the information about the critical points of real
functions defined on the manifold. Let us first introduce the definition of Morse function, and
then state the main results provided by Morse theory for the topological analysis of smooth man-
ifolds, such as surfaces. A basic reference for Morse theory is [ 141 ], while details about notions
of geometry and topology can be found, for example, in [ 104 ].

8.1 CRITICAL POINTS AND MORSE FUNCTIONS
Let M be a smooth compact n -dimensional manifold without boundary, and f WM!R a
smooth function defined on it. en, a point p of M is a critical point of f if we have

@x 1 .p/D0; @f

@f

@x 2 .p/D0;:::; @f @x n .p/D0;

with respect to a local coordinate system .x 1 ;:::;x n / about p . A real number is a critical value of
f if it is the image of a critical point. Points (values) which are not critical are said to be regular .
A critical point p is non-degenerate if the determinant of the Hessian matrix of f at p

@ 2 f
@x i @x j .p/

H f .p/D

is not zero; otherwise the critical point is degenerate . Figure 8.1 shows some examples of non-
degenerate and degenerate critical points. For a non-degenerate critical point p , the number of
negative eigenvalues of the Hessian H f .p/ of f at p determines the index of p , denoted by .p/ .
We say that f WM!R is a Morse function if all its critical points are non-degenerate.
A Morse function f is extremely simple near each non-degenerate critical point p . Indeed,
we can choose appropriate local coordinates .x 1 ;:::;x n / around p , in such a way that f has a
quadratic form representation: f.x 1 ;:::;x n /Df.p/

P P

.p/ n

i

1 x 2 i .
Intuitively, the index of a critical point is the number of independent directions around
the point in which the function decreases. For example, on a 2 -manifold, the indices of minima,
saddles, and maxima are 0 , 1 , and 2 , respectively.

iD1 x 2 i

C

D .p/ C