On the extendability of parallel sections of linear connections

Abstract

Let $\pi:E\to M$ be a vector bundle over a simply connected manifold and $\nabla$ a linear connection in $\pi$. Let $\sigma: U \rightarrow E$ be a $\nabla$-parallel section of $\pi$ defined on a connected open subset $U$ of $M$. We give sufficient conditions on $U$ in order to extend $\sigma$ to the whole $M$. We mainly concentrate to the case when $M$ is a $2$-dimensional simply connected manifold.