TY - RPRT
T1 - The curvature: a variational approach
Y1 - 2013
A1 - Andrei A. Agrachev
A1 - Davide Barilari
A1 - Luca Rizzi
KW - Crurvature, subriemannian metric, optimal control problem
AB - The 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7226
N1 - 88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections
on Finsler manifolds, slow growth distributions, Heisenberg group
U1 - 7260
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -
TY - JOUR
T1 - On 2-step, corank 2 nilpotent sub-Riemannian metrics
JF - SIAM J. Control Optim., 50 (2012) 559–582
Y1 - 2012
A1 - Davide Barilari
A1 - Ugo Boscain
A1 - Jean-Paul Gauthier
AB - In 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.
PB - Society for Industrial and Applied Mathematics
UR - http://hdl.handle.net/1963/6065
U1 - 5950
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - JOUR
T1 - A formula for Popp\'s volume in sub-Riemannian geometry
JF - Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
Y1 - 2012
A1 - Luca Rizzi
A1 - Davide Barilari
KW - subriemannian, volume, Popp, control
AB - For 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6501
N1 - 16 pages, minor revisions
U1 - 6446
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -
TY - JOUR
T1 - On the Hausdorff volume in sub-Riemannian geometry
JF - Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388
Y1 - 2012
A1 - Andrei A. Agrachev
A1 - Davide Barilari
A1 - Ugo Boscain
AB - For 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6454
U1 - 6399
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - RPRT
T1 - Introduction to Riemannian and sub-Riemannian geometry
Y1 - 2012
A1 - Andrei A. Agrachev
A1 - Davide Barilari
A1 - Ugo Boscain
PB - SISSA
UR - http://hdl.handle.net/1963/5877
U1 - 5747
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - JOUR
T1 - Sub-Riemannian structures on 3D Lie groups
JF - Journal of Dynamical and Control Systems. Volume 18, Issue 1, January 2012, Pages 21-44
Y1 - 2012
A1 - Andrei A. Agrachev
A1 - Davide Barilari
AB - We 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.

PB - SISSA
UR - http://hdl.handle.net/1963/6453
U1 - 6397
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -
TY - THES
T1 - Invariants, volumes and heat kernels in sub-Riemannian geometry
Y1 - 2011
A1 - Davide Barilari
KW - Sub-Riemannian geometry
AB - Sub-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]).
PB - SISSA
UR - http://hdl.handle.net/1963/6124
U1 - 6005
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -