A smooth function with certain special properties. Morse functions arise and are used in Morse theory.
Let be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary is a disconnected union (possibly empty) of manifolds and . A Morse function for the triple is a smooth (of Fréchet class ) function , (or for ), such that:
1) , ;
2) all critical points (cf. Critical point) of lie in and are non-degenerate;
For example, if is a proper function, that is, all sets , , are compact (possible only for ), then satisfies condition . A Morse function attains a (global) minimum on each connected component of . If is a finite-dimensional manifold, then for the set of Morse functions of class is a set of the second category (and, if is compact, even a dense open set) in the space of all functions
in the -topology.
|||M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)|
|||R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 (1963) pp. 299–340|
|||S. Smale, "Morse theory and a nonlinear generalization of the Dirichlet problem" Ann. of Math. , 80 (1964) pp. 382–396|
|[a1]||M. Goreski, R. MacPherson, "Stratified Morse theory" , Springer (1988)|
|[a2]||A. Wasserman, "Morse theory for -manifolds" Bull. Amer. Math. Soc. , 71 (1965) pp. 384–388|
|[a3]||A. Wasserman, "Equivariant differential topology" Topology , 8 (1969) pp. 127–150|
Morse function. M.M. PostnikovYu.B. Rudyak (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Morse_function&oldid=14633