Summary: Smooth functions statistics
V. I. Arnold
To describe the topological structure of a real smooth function one as-
sociates to it the graph, formed by the topological variety, whose points are
the connected components of the level hypersurface of the function.
For a Morse function f : Sn
R, n > 1, such a graph is a tree. Gener-
ically, it has T triple vertices, T + 2 endpoints, 2T + 2 vertices and 2T + 1
arrows.
Example 1. For the Elbrous mountain, with two maxima A and B, sepa-
rated by the saddle point C, the tree is
A
B
C
D
A
C
B
BA
D

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

Collections: Mathematics