{"paper":{"title":"How many geodesics join two points on a contact sub-Riemannian manifold?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OC","math.SG"],"primary_cat":"math.DG","authors_text":"Antonio Lerario, Luca Rizzi","submitted_at":"2014-05-16T20:00:14Z","abstract_excerpt":"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 $q=(x,z)$ satisfies: \\[ C_1\\frac{|z|}{\\|x\\|^2} + R_1 \\leq \\nu(q) \\leq C_2\\frac{|z|}{\\|x\\|^2} + R_2\\] for some constan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4294","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}