
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 is a normed vector space, , a subgroup, , of and a group-norm, , with a convexity property. We prove that with the metric , is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle over a Riemannian manifold, we prove that the triplet is a holonomic space, where is the holonomy group and is the length norm defined within. The topology on given by the 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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Analytic and geometric function theory
