01737nas a2200145 4500008004100000245005800041210005700099260001000156520126500166653006601431100002101497700001601518700002101534856003601555 2013 en d00aGeodesics and admissible-path spaces in Carnot Groups0 aGeodesics and admissiblepath spaces in Carnot Groups bSISSA3 aWe 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.10aCarnot group, Loop space, Betti number, Morse-Bott functional1 aAgrachev, Andrei1 aGentile, A.1 aLerario, Antonio uhttp://hdl.handle.net/1963/7228