A review of the tangent space in sub-Finsler geometry and applications to the failure of the CD condition

2024/04/25 by Magnabosco, Mattia, Rossi, Tommaso
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2404.16460

Abstract

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the CD condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.

Related

• Failure of the curvature-dimension condition in sub-Finsler manifolds
• Sub-Finsler geometry in dimension three
• Hopf-Rinow Theorem of sub-Finslerian geometry
• Convex Neighbourhoods and Complete Finsler Spaces
• Sub-twistor metrics