A.A. Agrachev, A. Gentile, A. Lerario, Geodesics and horizontal-path spaces in Carnot groups

Title: Geodesics and horizontal-path spaces in Carnot groups
Authors: A.A. Agrachev, A. Gentile, A. Lerario
Journal title: Geometry & Topology
Year: 2015
Volume: 19
Pages: 1569-1630
Abstract:

We study the topology of the space \Omega_{p} of admissible paths between two points e (the origin) and p on a step-two Carnot group G:

\Omega_{p} = \{\gamma \colon I \to G | \gamma admissible, \gamma(0) = e, \gamma(1) = p\}.

As it turns out, \Omega_{p} is homotopy equivalent to an infinite dimensional sphere and in particular it is contractible. The energy function:

J \colon \Omega_{p} \to \mathbb{R}

is defined by J(\gamma) = \frac{1}{2} \int_I {\Vert \dot{\gamma} \Vert}^2;critical points of this function are sub-Riemannian geodesics between e and p. We study the asymptotic of the number of geodesics and the topology of the sublevel sets:

{{\Omega}^s}_p = \{\gamma \in \Omega_{p} \| J(\gamma) \leq s\} as s \to \infty.

If p is not a vertical point in G, the number of geodesics joining e and p is bounded and the homology of {{\Omega}^s}_p stabilizes to zero for s large enough.

 

A completely different behavior is experienced for the generic vertical p. In this case we show that J is a Morse-Bott function: geodesics appear in isolated families (critical manifolds), indexed by their energy. Denoting by l the corank of the horizontal distribution on G, we prove that:

Card{Critical manifolds with energy less than s} \leq O(s)^l.

Despite this evidence, Morse-Bott inequalities b({{\Omega}^s}_p) \leq O(s)^l are far from being sharp and we show that the following stronger estimate holds:

b{{\Omega}^s}_p \leq O(s)^{l-1}. Thus each single Betti number b_i ({{\Omega}^s}_p) (i > 0) becomes eventually zero as s \to \infty, but the sum of all of them can possibly increase as fast as O(s)^{l-1}. In the case l = 2 we show that indeed

b{{\Omega}^s}_p = \tau (p)s + o(s) (l = 2). The leading order coefficient \tau(p) can be analytically computed using the structure constants of the Lie algebra of G.

 

Using a dilation procedure, reminiscent to the rescaling for Gromov-Hausdorff limits, we interpret these results as giving some local information on the geometry of G (e.g. we derive for l = 2 the rate of growth of the number of geodesics with bounded energy as p approaches e along a vertical direction).