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I 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 ∇ on E, any ∇-parallel section in E defined on U extends to a ∇-parallel section in E defined on M ? For simply connected manifolds M (among others) I 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 [1]. Furthermore, in case M is an open submanifold of R n , n ≥ 2, I prove that the complement of U in M , not required to be a submanifold now, can have arbitrarily large n-dimensional Lebesgue measure.

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