Legendre singularities of sub-Riemannian geodesics associated to Riemannian surfaces

Abstract

Let 𝑀 be a surface with a Riemannian metric and 𝑈𝑀 the unit tangent bundle over 𝑀 with the canonical contact sub-Riemannian structure 𝐷 ⊂ 𝑇(𝑈𝑀). In this paper, the complete local classification of singularities, under the Legendre projection 𝑈𝑀 → 𝑀, is given for sub-Riemannian geodesics of (𝑈𝑀, 𝐷). Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre projection from 𝑈𝑀 to the space of Riemannian geodesics on 𝑀. The duality on Legendre singularities is observed related to the pendulum motion.