Sub-Riemannian structures on 3D Lie groups

We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in terms of curvature invariants of a canonical connection. Moreover, 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

Comparison theorems for conjugate points in sub-Riemannian geometry

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet-Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.Comment: 33 pages, 5 figures, v2: minor revision, v3: minor revision, v4: minor revisions after publicatio

On Jacobi fields and canonical connection in sub-Riemannian geometry

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and we discuss some of its properties.Comment: 13 pages, (v2) minor corrections. Final version to appear on Archivum Mathematicu

Small time heat kernel asymptotics at the cut locus on surfaces of revolution

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.Comment: Accepted on Annales IHP - Analyse Non Lineair

On 2-step, corank 2 nilpotent sub-Riemannian metrics

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that 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

Invariants, volumes and heat kernels in sub-Riemannian geometry

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]) Very recently, it appeared in the eld of cognitive neuroscience to model the functional 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])

Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group

By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $\alpha_{1}$ and $\alpha_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($\alpha_{1}=\alpha_{2}$) and the non-isotropic cases ($\alpha_{1}\neq \alpha_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.Comment: 17 pages, 1 figur

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace-Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.Comment: 24 page