On the extendability of parallel sections of linear connections

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

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