01043nas a2200145 4500008004100000245004200041210003700083260001000120520060700130653006200737100002500799700002100824700001600845856003600861 2013 en d00aThe curvature: a variational approach0 acurvature a variational approach bSISSA3 aThe curvature discussed in this paper is a rather far going generalization of
the Riemannian sectional curvature. We define it for a wide class of optimal
control problems: a unified framework including geometric structures such as
Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special
attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces.
Our construction of the curvature is direct and naive, and it is similar to the
original approach of Riemann. Surprisingly, it works in a very general setting
and, in particular, for all sub-Riemannian spaces.10aCrurvature, subriemannian metric, optimal control problem1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aRizzi, Luca uhttp://hdl.handle.net/1963/722600934nas a2200133 4500008004100000245005700041210005100098260005100149520050200200100002100702700001700723700002400740856003600764 2012 en d00aOn 2-step, corank 2 nilpotent sub-Riemannian metrics0 a2step corank 2 nilpotent subRiemannian metrics bSociety for Industrial and Applied Mathematics3 aIn this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.1 aBarilari, Davide1 aBoscain, Ugo1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/606501179nas a2200133 4500008004100000245006000041210005500101260001000156520076500166653004100931100001600972700002100988856003601009 2012 en d00aA formula for Popp\'s volume in sub-Riemannian geometry0 aformula for Popps volume in subRiemannian geometry bSISSA3 aFor an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.10asubriemannian, volume, Popp, control1 aRizzi, Luca1 aBarilari, Davide uhttp://hdl.handle.net/1963/650101194nas a2200133 4500008004100000245005500041210004700096260001000143520080800153100002500961700002100986700001701007856003601024 2012 en d00aOn the Hausdorff volume in sub-Riemannian geometry0 aHausdorff volume in subRiemannian geometry bSISSA3 aFor a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aBoscain, Ugo uhttp://hdl.handle.net/1963/645400389nas a2200121 4500008004100000245005900041210005800100260001000158100002500168700002100193700001700214856003600231 2012 en d00aIntroduction to Riemannian and sub-Riemannian geometry0 aIntroduction to Riemannian and subRiemannian geometry bSISSA1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aBoscain, Ugo uhttp://hdl.handle.net/1963/587700752nas a2200121 4500008004100000245004700041210004600088260001000134520040400144100002500548700002100573856003600594 2012 en d00aSub-Riemannian structures on 3D Lie groups0 aSubRiemannian structures on 3D Lie groups bSISSA3 aWe give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(\mathbb{R})\times S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.

1 aAgrachev, Andrei, A.1 aBarilari, Davide uhttp://hdl.handle.net/1963/645301397nas a2200121 4500008004100000245006800041210006600109260001000175520100500185653002801190100002101218856003601239 2011 en d00aInvariants, volumes and heat kernels in sub-Riemannian geometry0 aInvariants volumes and heat kernels in subRiemannian geometry bSISSA3 aSub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])\\r\\nVery recently, it appeared in the eld of cognitive neuroscience to model the\\r\\nfunctional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]).10aSub-Riemannian geometry1 aBarilari, Davide uhttp://hdl.handle.net/1963/6124