In General > s.a. connections; crystals;
homology types [Morse homology];
* Idea: A modern version of the calculus of variations, which uses infinite-dimensional manifolds, their points being the geodesics of a given manifold.
* Applications: Studying symmetries in crystals (study symmetry breaking by minimizing the potential energy).
@ General references: Morse 64; Milnor 73; Rassias 92; Matsumoto 01; Harvey & Lawson AM(01) [based on de Rham-Federer theory of currents]; Nicolaescu 07; Katz 14 [manifolds with boundary]; Knudson 15 [smooth and discrete Morse theory].
@ Morse index theorem: Rezende LMP(98) [proof]; Piccione & Tausk JMP(99), Top(02) [semi-Riemannian geometry].
@ In general relativity: Woodhouse CMP(76) [and spacetime topology]; Giannoni et al JGP(00) [particles]; > s.a. Fermat's Principle.
@ Related topics: Floer BAMS(87); Ghrist et al m.DS/01 [on spaces of braids, and Lagrangian dynamics].
@ Generalizations: Goresky & MacPherson 88 [stratified]; Perlick JMP(95) [infinite-dimensional]; Minian T&A(12)-a1007 [discrete, for posets].
> Online resources: see MathWorld page; Wikipedia page.
* Idea: Once a Morse function has been defined on a manifold, information about its topology can be deduced from its critical elements.
$ Def: Given a smooth cobordism between M and N, a Morse function is a function f with nowhere vanishing gradient except at isolated points, where the Hessian is non-degenerate.
* In general relativity: It can be used as time to define a Lorentzian metric starting from a Riemannian one; The Lorentzian metric will be degenerate where f has vanishing gradient, but it will not have closed timelike curves.
@ Discrete: Lewiner et al CG(03) [2D]; Chari & Joswig DM(05) [complex of discrete Morse functions of a fixed simplicial complex]; Ayala et al T&A(09) [Morse theory and topology of graphs]; Sawicki JPA(12) [for graph configuration spaces, and quantum statistics for particles on networks]; Bloch DM(13) [polyhedral representation].
$ Def: If f : M → \(\mathbb R\), with M a compact n-dimensional differentiable manifold, ck the number of non-degenerate critical points with index k, and Rk(M) the k-th Betti number, then
Rk(M) – Rk–1(M) + ... ± R0(M) ≤ ck – ck+1 + ... ± c0 ,
with equality if k = n.
* Corollary: For all integers k, ck > Rk(M).
Morse Vector Field
$ Def: Given a manifold M with boundary consisting of two disjoint components, ∂M = M0 ∪ M1, a Morse vector field is a vector field which points inwards on M and outwards on M.
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