Curvature exponent and geodesic dimension on Sard-regular Carnot groups

Abstract

In this study, we characterize the geodesic dimension NGEO{N}_{{\rm{GEO}}} and give a new lower bound to the curvature exponent NCE{N}_{{\rm{CE}}} on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which NCE>NGEO{N}_{{\rm{CE}}}\gt {N}_{{\rm{GEO}}}; this answers a question posed by Rizzi (Measure contraction properties of Carnot groups. Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 60, 20).

Keywords

• carnot groups
• curvature exponent
• geodesic dimension
• sub-riemannian geometry
• 53c17
• 53c23