Metric Lines in the Special Euclidean group on the plane

2024/08/25 by Wang, Y., Ku, S., Bravo-Doddoli, A.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.13873

Abstract

The Special Euclidean group on the plane SE(2) has the left-invariant sub-Riemannian structure. Every sub-Riemannian manifold possesses a Hamiltonian function governing the sub-Riemannian geodesic flow. Two natural questions are: What are the necessary conditions for periodic sub-Riemannian geodesics? What geodesics are the metric lines in SE(2)? We answer both questions, and our method for the second is an alternative proof using the Hamilton-Jacobi theory.

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