Extendability of parallel sections in vector bundles

Abstract

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 (2014). 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.