Analysis seminar - Metric lines in Carnot groups

Abstract:

 Within the class of nilpotent Lie groups, Carnot groups occupy a particularly distinguished place, with the Heisenberg group serving as the most familiar example. Carnot groups naturally carry the structure of sub-Riemannian manifolds, where a geodesic is understood as a locally arc-length–minimizing curve. This raises a fundamental question: under what conditions does a geodesic extend to a global minimizer? Such curves are called metric lines, or equivalently, isometric embeddings of the real line. This talk is devoted to presenting the results and ideas that led me to formulate a conjecture on the classification of metric lines in Carnot groups, along with recent progress toward its resolution.