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 [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.

Uniform Stability of Linear Evolution Equations with Applications to Parallel Transports

“…The following definition stems from a previous paper of mine [5]. In case M is a, say simply connected 7 , second-countable Hausdorff manifold of dimension ≥ 2 and F is a closed submanifold, boundary allowed, of class C 1 of M , we know [5] that F is negligible in M for all connections of class C 0 if and only if M \ F is dense and connected in M . This result relies heavily on the fact that when F ⊂ M is a closed C 1 submanifold, with possible boundary, F can be locally flattened by means of a diffeomorphism.…”