Morse Theory| Morse Theory |
In General > s.a. connections; crystals;
homology types [Morse homology]; jacobi metric.
* 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];
Kirwan & Penington a1906 [without non-degeneracy assumptions].
@ Morse index theorem: Rezende LMP(98) [proof];
Piccione & Tausk JMP(99),
Top(02) [semi-Riemannian geometry].
@ 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].
@ And physics: Woodhouse CMP(76) [and spacetime topology];
Giannoni et al JGP(00) [particles];
> s.a. Fermat's Principle;
topological field theories.
> Online resources:
see MathWorld page;
Wikipedia page.
Morse Function
* 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].
Morse Inequalities
$ 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, c k >
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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