01188nas a2200157   4500008004100000245005800041210005700099260003700156300001600193490000700209520065100216100002500867700002400892700002100916856009300937       2015                        eng d00aGeodesics and horizontal-path spaces in Carnot groups0 aGeodesics and horizontalpath spaces in Carnot groups  bMathematical Sciences Publishers  a1569–16300 v193 a<p>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.</p>1 aAgrachev, Andrei, A.1 aGentile, Alessandro1 aLerario, Antonio  uhttps://www.math.sissa.it/publication/geodesics-and-horizontal-path-spaces-carnot-groups