The geometry of Riemannian curvature radii

TL;DR

This paper investigates the geometric structures linked to curvature radii of curves on Riemannian manifolds, revealing natural sub-Riemannian structures and intrinsic geometric information encoded by specific vector fields.

Contribution

It introduces a novel geometric framework connecting curvature radii with sub-Riemannian manifolds and identifies key vector fields encoding intrinsic manifold geometry.

Findings

Existence of sub-Riemannian structures associated with curvature radii

Identification of global vector fields encoding intrinsic geometry

Insights into the geometric properties of curvature radii on Riemannian manifolds

Abstract

In this paper we explore the geometric structures associated with curvature radii of curves with values on a Riemannian manifold $(M, g)$. We show the existence of sub-Riemannian manifolds naturally associated with the curvature radii and we investigate their properties. The main character of our construction is a pair of global vector fields $f_1, f_2$, which encodes intrinsic information about the geometry of $(M, g)$.