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Lower semicontinuity of restricted holonomy groups

Linus Götzfried

TL;DR

This work investigates how holonomy groups of metric connections behave under limits, proving a monotony principle for $C^0$-convergent sequences of Lipschitz metric connections: if each $ ext{Hol}_x( abla^{k})$ lies in a closed group $H$, then the limit $ ext{Hol}_x( abla^{g})$ lies in $H$ after an appropriate basis. The proof uses a nonstandard embedding $oldsymbol{\phi}_h: ext{SO}(l) o ext{SO}(l)_h$, together with a compactness/continuity argument for parallel transport along loops and a Cantor–Bernstein-type step to extract a limiting conjugation $U$. As a key application to Riemannian geometry, the conjugacy class of the restricted holonomy $ ext{Hol}_x^0(g)$ is shown to be lower semicontinuous with respect to the $C^1$-topology on the space of metrics, implying that the holonomy can only become more special in the limit. The paper also discusses corollaries about closedness of holonomy-constrained metric sets and situates the results relative to prior work, noting potential generalizations and limitations (e.g., issues with full holonomy and non-Lorentzian settings).

Abstract

A result on the monotony of holonomy groups of metric connections in limits is proven. The main outcome is that given a sequence of metric connections that are Lipschitz continuous and converging in $C^0$, if the connections in the sequence all have holonomy contained in a closed group $H$, then also the limit connection has holonomy contained in $H$. In particular, this finds an application for Riemannian holonomy groups, where it is proven that the map assigning to a Riemannian metric the conjugacy class of its restricted holonomy group is lower semicontinuous with respect to the $C^1$-topology on the space of metrics.

Lower semicontinuity of restricted holonomy groups

TL;DR

This work investigates how holonomy groups of metric connections behave under limits, proving a monotony principle for -convergent sequences of Lipschitz metric connections: if each lies in a closed group , then the limit lies in after an appropriate basis. The proof uses a nonstandard embedding , together with a compactness/continuity argument for parallel transport along loops and a Cantor–Bernstein-type step to extract a limiting conjugation . As a key application to Riemannian geometry, the conjugacy class of the restricted holonomy is shown to be lower semicontinuous with respect to the -topology on the space of metrics, implying that the holonomy can only become more special in the limit. The paper also discusses corollaries about closedness of holonomy-constrained metric sets and situates the results relative to prior work, noting potential generalizations and limitations (e.g., issues with full holonomy and non-Lorentzian settings).

Abstract

A result on the monotony of holonomy groups of metric connections in limits is proven. The main outcome is that given a sequence of metric connections that are Lipschitz continuous and converging in , if the connections in the sequence all have holonomy contained in a closed group , then also the limit connection has holonomy contained in . In particular, this finds an application for Riemannian holonomy groups, where it is proven that the map assigning to a Riemannian metric the conjugacy class of its restricted holonomy group is lower semicontinuous with respect to the -topology on the space of metrics.
Paper Structure (3 sections, 6 theorems, 4 equations)

This paper contains 3 sections, 6 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Monotony of the holonomy group in limits
  3. Application to Riemannian holonomy groups

Key Result

Theorem 2.4

Assume Situation sit:HolonomyLimit and let $H\subseteq\mathrm{SO}(l)$ be closed. Moreover, for $k\in\mathbb{N}$, let $g_k\in C^1(M,\mathrm{S}^2 E^*)$ be a bundle metric on $E$ and $\nabla^{k}$ be a connection with locally Lipschitz continuous coefficients compatible with $g_k$, such that $g_k|_x\to

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.3
  • Theorem 2.4
  • proof : Proof of Theorem \ref{['thm:HolonomyLimit']}
  • Remark 2.5
  • Definition 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Definition 3.4
  • Theorem 3.5
  • ...and 8 more