Open Access
2015 Geodesics and horizontal-path spaces in Carnot groups
Andrei A Agrachev, Alessandro Gentile, Antonio Lerario
Geom. Topol. 19(3): 1569-1630 (2015). DOI: 10.2140/gt.2015.19.1569


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.


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Andrei A Agrachev. Alessandro Gentile. Antonio Lerario. "Geodesics and horizontal-path spaces in Carnot groups." Geom. Topol. 19 (3) 1569 - 1630, 2015.


Received: 22 January 2014; Revised: 16 August 2014; Accepted: 30 August 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1333.53043
MathSciNet: MR3352244
Digital Object Identifier: 10.2140/gt.2015.19.1569

Primary: 53C17
Secondary: 37J60 , 58E05

Keywords: Carnot groups , loop spaces , Morse–Bott theory , sub-Riemannian geometry

Rights: Copyright © 2015 Mathematical Sciences Publishers

Vol.19 • No. 3 • 2015
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