[Paper Review] Holonomic Spaces

Pedro Solórzano|arXiv (Cornell University)|Apr 9, 2010

Geometric Analysis and Curvature Flows14 references4 citations

TL;DR

This paper introduces holonomic spaces—normed vector spaces equipped with a subgroup of isometries and a convex group norm—to define a metric that makes the space locally isometric to a Euclidean ball. It proves the holonomy radius of a Riemannian manifold is positive and applies the framework to study Gromov-Hausdorff convergence.

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.

Motivation & Objective

• To formalize a geometric structure—holonomic spaces—unifying isometry groups and group norms in normed vector spaces.
• To define a metric on vector spaces that induces local Euclidean structure, enabling geometric analysis of holonomy groups.
• To establish the positivity of the holonomy radius in Riemannian manifolds using the new metric framework.
• To apply the framework to the study of Gromov-Hausdorff convergence of Riemannian manifolds.
• To show that the $L_p$-induced topology on the holonomy group is finer than the subspace topology while preserving key geometric properties.

Proposed method

• Define a holonomic space $(V, H, L)$ as a normed vector space $V$, a subgroup $H$ of isometries of $V$, and a convex group norm $L$.
• Construct the metric $d_L(u,v) = \inf_{a \in H} \left\{ \sqrt{L^2(a) + \|u - a v\|^2} \right\}$ on $V$.
• Prove that this metric makes $V$ locally isometric to a Euclidean ball via the local structure of $H$ and convexity of $L$.
• Apply the framework to vector bundles with Sasaki-type metrics, showing $(E_p, \text{Hol}_p, L_p)$ forms a holonomic space.
• Define the length norm $L_p$ on the holonomy group $\text{Hol}_p$ and show the induced topology is finer than the subspace topology.
• Use the metric structure to define and prove the positivity of the holonomy radius for Riemannian manifolds.

Experimental results

Research questions

• RQ1 Can a metric be defined on a normed vector space with a group of isometries and a group norm such that the space is locally Euclidean?
• RQ2 How does the $L_p$-induced topology on the holonomy group compare to the subspace topology in the context of vector bundles?
• RQ3 What geometric invariants can be derived from the holonomic space structure in Riemannian geometry?
• RQ4 Does the holonomy radius of a Riemannian manifold remain positive under the proposed metric framework?
• RQ5 To what extent can this framework be applied to the Gromov-Hausdorff convergence of Riemannian manifolds?

Key findings

• The metric $d_L(u,v) = \inf_{a \in H} \left\{ \sqrt{L^2(a) + \|u - a v\|^2} \right\}$ endows the vector space $V$ with a structure that is locally isometric to a Euclidean ball.
• For a vector bundle with a Sasaki-type metric, the triple $(E_p, \text{Hol}_p, L_p)$ forms a holonomic space, where $\text{Hol}_p$ is the holonomy group and $L_p$ is the length norm.
• The topology on $\text{Hol}_p$ induced by $L_p$ is strictly finer than the subspace topology while retaining essential geometric properties.
• The holonomy radius of a Riemannian manifold is proven to be positive using the holonomic space framework.
• The framework provides a new analytical tool for studying Gromov-Hausdorff convergence by embedding geometric constraints into the holonomy structure.
• The convexity property of the group norm $L$ ensures the metric $d_L$ satisfies the necessary geometric axioms for a well-behaved distance function.