1,580 research outputs found
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 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
Sharp measure contraction property for generalized H-type Carnot groups
We prove that H-type Carnot groups of rank and dimension satisfy the
if and only if and . The latter
integer coincides with the geodesic dimension of the Carnot group. The same
result holds true for the larger class of generalized H-type Carnot groups
introduced in this paper, and for which we compute explicitly the optimal
synthesis. This constitutes the largest class of Carnot groups for which the
curvature exponent coincides with the geodesic dimension. We stress that
generalized H-type Carnot groups have step 2, include all corank 1 groups and,
in general, admit abnormal minimizing curves.
As a corollary, we prove the absolute continuity of the Wasserstein geodesics
for the quadratic cost on all generalized H-type Carnot groups.Comment: 18 pages. This article extends the results of arXiv:1510.05960. v2:
revised and improved version. v3: final version, to appear in Commun.
Contemp. Mat
- …