Lerario: Geodesics and horizontal-path spaces in Carnot groups, Geometry & Topology

Abstract. We study the topology of the space Ωp of horizontal paths between two points e (the origin) and p on a step-two Carnot group G: Ωp = {γ: I → G | γ horizontal, γ(0) = e, γ(1) = p}. As it turns out, Ωp is homotopy equivalent to an infinite dimensional sphere and in particular it is contractible. The energy function