Geodesics and horizontal-path spaces in Carnot groups

Abstract

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.