@article {agrachev2015geodesics,
	title = {Geodesics and horizontal-path spaces in Carnot groups},
	journal = {Geometry \& Topology},
	volume = {19},
	number = {3},
	year = {2015},
	pages = {1569{\textendash}1630},
	publisher = {Mathematical Sciences Publishers},
	abstract = {<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>},
	doi = {10.2140/gt.2015.19.1569},
	author = {Andrei A. Agrachev and Alessandro Gentile and Antonio Lerario}
}
@article {2011,
	title = {Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds},
	number = {arXiv:0903.2550;},
	year = {2011},
	note = {This is a revised extended version that contains new results.},
	publisher = {SISSA},
	url = {http://hdl.handle.net/1963/6507},
	author = {Andrei A. Agrachev and Paul Lee}
}
@article {2011,
	title = {The geometry of Maximum Principle},
	journal = {Proceedings of the Steklov Institute of mathematics. vol. 273 (2011), page: 5-27 ; ISSN: 0081-5438},
	number = {Proceedings of the Steklov Institute of mathematics;v.273},
	year = {2011},
	abstract = {An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.},
	url = {http://hdl.handle.net/1963/6456},
	author = {Andrei A. Agrachev and Revaz Gamkrelidze}
}
@article {2008,
	title = {A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds},
	journal = {Discrete Contin. Dyn. Syst. 20 (2008) 801-822},
	number = {SISSA;55/2006/M},
	year = {2008},
	abstract = {We 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.},
	doi = {10.3934/dcds.2008.20.801},
	url = {http://hdl.handle.net/1963/1869},
	author = {Andrei A. Agrachev and Ugo Boscain and Mario Sigalotti}
}
@article {2002,
	title = {Geometry of Jacobi Curves I},
	journal = {J. Dynam. Control Systems 8 (2002) 93-140},
	number = {SISSA;75/2001/M},
	year = {2002},
	publisher = {Springer},
	abstract = {Jacobi 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\\\".},
	doi = {10.1023/A:1013904801414},
	url = {http://hdl.handle.net/1963/3110},
	author = {Andrei A. Agrachev and Igor Zelenko}
}
@article {2002,
	title = {Geometry of Jacobi curves II},
	journal = {J. Dynam. Control Systems 8 (2002), no. 2, 167--215},
	number = {SISSA;18/2002/M},
	year = {2002},
	publisher = {SISSA Library},
	doi = {10.1023/A:1015317426164},
	url = {http://hdl.handle.net/1963/1589},
	author = {Andrei A. Agrachev and Igor Zelenko}
}
