Extendability of parallel sections in vector bundles

TL;DR

This paper investigates the conditions under which parallel sections of vector bundles over manifolds can be extended from open subsets to the entire manifold, providing characterizations for simply connected manifolds and measure-theoretic results for open subsets of Euclidean spaces.

Contribution

It characterizes open subsets allowing extension of parallel sections on simply connected manifolds and shows the measure-theoretic size of non-extendable sets in Euclidean spaces.

Findings

For simply connected manifolds, such open sets are complements of $C^1$ submanifolds.

In Euclidean spaces, the complement of such sets can have arbitrarily large Lebesgue measure.

Provides a partial answer to a problem posed by Di Scala and Manno.

Abstract

We address the following question: Given a differentiable manifold $M$ what are the open subsets $U$ of $M$ such that, for all vector bundles $E$ over $M$ and all linear connections $\nabla$ on $E$, any $\nabla$-parallel section in $E$ defined on $U$ extends to a $\nabla$-parallel section in $E$ defined on $M$? For simply connected manifolds $M$ (among others) we describe the entirety of all such sets $U$ which are, in addition, the complement of a $C^1$ submanifold (boundary allowed) of $M$; this delivers a partial positive answer to a problem posed by Antonio J. Di Scala and Gianni Manno. Furthermore, in case $M$ is an open submanifold of $\mathbb R^n$, $2 \leq n$, we prove that the complement of $U$ in $M$, not required to be a submanifold now, can have arbitrarily large $n$-dimensional Lebesgue measure.