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Geodesics and horizontal-path spaces in Carnot groups

Andrei A Agrachev, Alessandro Gentile and Antonio Lerario

Geometry & Topology 19 (2015) 1569–1630

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

Loop spaces, Carnot groups, Morse–Bott theory, sub-Riemannian geometry
Mathematical Subject Classification 2010
Primary: 53C17
Secondary: 37J60, 58E05
Received: 22 January 2014
Revised: 16 August 2014
Accepted: 30 August 2014
Published: 21 May 2015
Proposed: Yasha Eliashberg
Seconded: Dmitri Burago, Leonid Polterovich
Andrei A Agrachev
SISSA & IM SO RAS I-34136 Trieste
Alessandro Gentile
I-34136 Trieste
Antonio Lerario
Institut Camille Jordan
69622 Villeurbanne