Curvature exponent and geodesic dimension on Sard-regular Carnot groups

TL;DR

This paper investigates the geometric properties of Sard-regular Carnot groups by characterizing the geodesic dimension and providing bounds for the curvature exponent, including a counterexample that challenges previous assumptions.

Contribution

It introduces a new characterization of the geodesic dimension and establishes a lower bound for the curvature exponent, along with a counterexample demonstrating that the curvature exponent can exceed the geodesic dimension.

Findings

Characterization of the geodesic dimension $N_{GEO}$

New lower bound for the curvature exponent $N_{CE}$

Counterexample of a step-two Carnot group where $N_{CE} > N_{GEO}$

Abstract

In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which $N_{CE} > N_{GEO}$: this answers a question posed by Rizzi in arXiv:1510.05960v4 [math.MG].