# 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} }

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

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