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Geodesics and admissible-path spaces in Carnot Groups

TitleGeodesics and admissible-path spaces in Carnot Groups
Publication TypePreprint
AuthorsAgrachev, A, Gentile, A, Lerario, A
Document NumberarXiv:1311.6727;
Carnot group, Loop space, Betti number, Morse-Bott functional

We study the topology of admissible-loop spaces on a step-two Carnot group G.
We use a Morse-Bott theory argument to study the structure and the number of
geodesics on G connecting the origin with a 'vertical' point (geodesics are
critical points of the 'Energy' functional, defined on the loop space). These
geodesics typically appear in families (critical manifolds). Letting the energy
grow, we obtain an upper bound on the number of critical manifolds with energy
bounded by s: this upper bound is polynomial in s of degree l (the corank of
the distribution). Despite this evidence, we show that Morse-Bott inequalities
are far from sharp: the topology (i.e. the sum of the Betti numbers) of the
loop space filtered by the energy grows at most as a polynomial in s of degree
l-1. In the limit for s at infinity, all Betti numbers (except the zeroth) must
actually vanish: the admissible-loop space is contractible. In the case the
corank l=2 we compute exactly the leading coefficient of the sum of the Betti
numbers of the admissible-loop space with energy less than s. This coefficient
is expressed by an integral on the unit circle depending only on the
coordinates of the final point and the structure constants of the Lie algebra
of G.

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