2014/07/07 by Tim Kirschner, Kirschner, Tim
Mathematics · #53C05 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.DG #msc:53C05
paper · pdf · doi:10.48550/arxiv.1407.1727
Improved presentation
arxiv created 2015/08/26 · arxiv updated 2015/08/27
We 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) we 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. Furthermore, in case M is an open submanifold of \mathbb Rn, 2 ≤ n, we prove that the complement of U in M, not required to be a submanifold now, can have arbitrarily large n-dimensional Lebesgue measure.