Holonomic Spaces

Abstract

A holonomic space (V, H, L) is a normed vector space, V , a subgroup, H, of Aut(V, • ) and a group-norm, L, with a convexity property. We prove that with the metric d L (u, v) = inf a∈H L 2 (a) + u − av 2 , V is a metric space which is locally isometric to a Euclidean ball. Given a Sasakitype metric on a vector bundle E over a Riemannian manifold, we prove that the triplet (Ep, Holp, Lp) is a holonomic space, where Holp is the holonomy group and Lp is the length norm defined within. The topology on Holp given by the Lp 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.