Extendability of parallel sections in vector bundles

For simply connected manifolds M (among others) I describe the entirety of all such sets U which are, in addition, the complement of a C1 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 Rn, 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.