601 research outputs found
A contact covariant approach to optimal control with applications to sub-Riemannian geometry
We discuss contact geometry naturally related with optimal control problems
(and Pontryagin Maximum Principle). We explore and expand the observations of
[Ohsawa, 2015], providing simple and elegant characterizations of normal and
abnormal sub-Riemannian extremals.Comment: A small correction in the statement and proof of Thm 6.15. Watch our
publication: https://youtu.be/V04N9X3NxYA and https://youtu.be/jghdRK2IaU
Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds
For a fat sub-Riemannian structure, we introduce three canonical Ricci
curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we
prove comparison theorems for conjugate lengths, Bonnet-Myers type results and
Laplacian comparison theorems for the intrinsic sub-Laplacian.
As an application, we consider the sub-Riemannian structure of -Sasakian
manifolds, for which we provide explicit curvature formulas. We prove that any
complete -Sasakian structure of dimension , with , has
sub-Riemannian diameter bounded by . When , a similar statement holds
under additional Ricci bounds. These results are sharp for the natural
sub-Riemannian structure on of the quaternionic Hopf
fibrations: \begin{equation*} \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3}
\to \mathbb{HP}^d, \end{equation*} whose exact sub-Riemannian diameter is
, for all .Comment: 34 pages, v2: fixed and clarified the proof of Theorem 7 and some
typos, v3: final version, to appear on Journal of the Institute of
Mathematics of Jussie
Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds
Measure contraction property is one of the possible generalizations of Ricci
curvature bound to more general metric measure spaces. In this paper, we
discover sufficient conditions for a three dimensional contact subriemannian
manifold to satisfy this property.Comment: 49 page
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
The rolling problem: overview and challenges
In the present paper we give a historical account -ranging from classical to
modern results- of the problem of rolling two Riemannian manifolds one on the
other, with the restrictions that they cannot instantaneously slip or spin one
with respect to the other. On the way we show how this problem has profited
from the development of intrinsic Riemannian geometry, from geometric control
theory and sub-Riemannian geometry. We also mention how other areas -such as
robotics and interpolation theory- have employed the rolling model.Comment: 20 page
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
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