Infinite geodesics and isometric embeddings in Carnot groups of step 2
In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle.
The author has been partially supported by the Vilho, Yrjö and Kalle Väisälä Foundation, by the Academy of Finland (grant 288501 “Geometry of subRiemannian groups”), and by the European Research Council (ERC Starting Grant 713998 GeoMeG “Geometry of Metric Groups”).
The author wishes to thank Enrico Le Donne and Yuri Sachkov for helpful discussions on the control viewpoint to infinite geodesics that paved the way to the conclusion of the main proof. The author also wishes to thank Ville Kivioja for his help during the outset of the project in the study of the known results and their abstractions.