Geodesics and horizontal-path spaces in Carnot groups

Agrachev, Andrei A; Gentile, Alessandro; Lerario, Antonio

doi:10.2140/gt.2015.19.1569

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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

DOI: 10.2140/gt.2015.19.1569

Abstract

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.

Keywords

Loop spaces, Carnot groups, Morse–Bott theory, sub-Riemannian geometry

Mathematical Subject Classification 2010

Primary: 53C17

Secondary: 37J60, 58E05

References

Publication

Received: 22 January 2014

Revised: 16 August 2014

Accepted: 30 August 2014

Published: 21 May 2015

Proposed: Yasha Eliashberg

Seconded: Dmitri Burago, Leonid Polterovich

Authors

Andrei A Agrachev
SISSA & IM SO RAS I-34136 Trieste Italy

Alessandro Gentile
Sissa I-34136 Trieste Italy

Antonio Lerario
Institut Camille Jordan 69622 Villeurbanne France

Volume 19, issue 3 (2015)