# How many geodesics join two points on a contact sub-Riemannian manifold?

@article{Lerrio2017HowMG,
title={How many geodesics join two points on a contact sub-Riemannian manifold?},
author={Antonio Marcondes Ler{\'a}rio and Luca Rizzi},
journal={Journal of Symplectic Geometry},
year={2017},
volume={15},
pages={247-305}
}
• Published 2017
• Mathematics
• Journal of Symplectic Geometry
We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z) \in \mathbb{R}^{2n} \times \mathbb{R}$ centered at $p$. In this case we prove that for the generic $q$ the number of geodesics $\nu(q)$ between $p$ and $q=(x,z… Expand 6 Citations #### Figures from this paper On the cut locus of free, step two Carnot groups • Mathematics • 2016 In this note, we study the cut locus of the free, step two Carnot groups$\mathbb{G}_k$with$k$generators, equipped with their left-invariant Carnot-Carath\'eodory metric. In particular, weExpand Sharp measure contraction property for generalized H-type Carnot groups • Mathematics • 2017 We prove that H-type Carnot groups of rank$k$and dimension$n$satisfy the$\mathrm{MCP}(K,N)$if and only if$K\leq 0$and$N \geq k+3(n-k)$. The latter integer coincides with the geodesicExpand Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry • Mathematics • 2017 On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ∆ω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, asExpand Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry • Mathematics • 2017 We discuss homotopy properties of endpoint maps for horizontal path spaces, i.e. spaces of curves on a manifold M whose velocities are constrained to a subbundle Δ ⊂ TM in a nonholonomic way. WeExpand Short Geodesics Losing Optimality in Contact Sub-Riemannian Manifolds and Stability of the 5-Dimensional Caustic An approximation of the sub-Riemannian Hamiltonian flow is computed and a geometric invariant is introduced to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. Expand Besicovitch Covering Property on graded groups and applications to measure differentiation • Mathematics • Journal für die reine und angewandte Mathematik (Crelles Journal) • 2019 We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admitsExpand #### References SHOWING 1-10 OF 23 REFERENCES Length of Geodesics and Quantitative Morse Theory on Loop Spaces • Mathematics • 2013 Let Mn be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points $${p,q \in M^n}$$ and every positive integer k there are at least kExpand On the Hausdorff volume in sub-Riemannian geometry • Mathematics • 2012 For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball inExpand Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry • Mathematics • 2015 We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some$W^{1,p}\$ topology on the space of trajectories, forExpand
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