Left-invariant sub-Riemannian structures on a five-dimensional two-step nilpotent Lie group: isometries, classification, and geodesics

Abstract

In this dissertation we study a five-dimensional two-step nilpotent matrix Lie group. Some basic group properties are investigated. The structure of the Lie algebra’s subspaces is investigated; a complete set of scalar invariants is given for the Lie algebra’s subspace structure. Following this, we classify the left-invariant sub-Riemannian structures on this Lie group up to isometry. The normal geodesics of the rank three left-invariant sub-Riemannian structure are determined as an illustrative case.