Holonomic Spaces

TL;DR

This paper introduces the concept of holonomic spaces, showing they are locally Euclidean, and applies this to Riemannian manifolds, establishing properties of holonomy groups and a positive holonomy radius relevant for geometric convergence.

Contribution

It defines holonomic spaces with a new metric structure, relates them to holonomy groups in Riemannian geometry, and introduces the holonomy radius with implications for Gromov-Hausdorff convergence.

Findings

Holonomic spaces are locally isometric to Euclidean balls.

The holonomy group with the length norm has a finer topology.

The holonomy radius of a Riemannian manifold is positive.

Abstract

A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$, of $Aut(V, \|\cdot\|)$ and a group-norm, $L$, with a convexity property. We prove that with the metric $d_L(u,v)=\inf_{a\in H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$, $V$ is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle $E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than the subspace topology while still preserving many desirable properties. Using these notions, we introduce the notion of holonomy radius for a Riemannian manifold and prove it is positive. These results are applicable to the Gromov-Hausdorff convergence of Riemannian manifolds.