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; Harvey & Lawson AM(01)
[based on de Rham-Federer theory of currents].
@ 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].
@ Generalizations: Goresky & MacPherson 88 [stratified]; Perlick JMP(95)
[infinite-dimensional].
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].
Morse Inequalities
$ Def: If f : M → 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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27 jun 2008