# 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).