Extendability of parallel sections in vector bundles

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.