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-groups00398nas a2200109   4500008004100000245009300041210006900134260001000203100002500213700001400238856003600252       2011                        en  d00aGeneralized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds0 aGeneralized Ricci Curvature Bounds for Three Dimensional Contact  bSISSA1 aAgrachev, Andrei, A.1 aLee, Paul  uhttp://hdl.handle.net/1963/650700442nas a2200109   4500008004100000245003800041210003400079520013500113100002500248700002300273856003600296       2011                        en  d00aThe geometry of Maximum Principle0 ageometry of Maximum Principle3 aAn invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.1 aAgrachev, Andrei, A.1 aGamkrelidze, Revaz  uhttp://hdl.handle.net/1963/645601551nas a2200121   4500008004300000245007900043210006900122520113900191100002501330700001701355700002101372856003601393       2008                        en_Ud  00aA Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds0 aGaussBonnetlike formula on twodimensional almostRiemannian manif3 aWe consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aSigalotti, Mario  uhttp://hdl.handle.net/1963/186901353nas a2200121   4500008004300000245003200043210003200075260001300107520103200120100002501152700001801177856003601195       2002                        en_Ud  00aGeometry of Jacobi Curves I0 aGeometry of Jacobi Curves I  bSpringer3 aJacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".1 aAgrachev, Andrei, A.1 aZelenko, Igor  uhttp://hdl.handle.net/1963/311000314nas a2200109   4500008004100000245003300041210003300074260001800107100002500125700001800150856003600168       2002                        en  d00aGeometry of Jacobi curves II0 aGeometry of Jacobi curves II  bSISSA Library1 aAgrachev, Andrei, A.1 aZelenko, Igor  uhttp://hdl.handle.net/1963/1589