Extendability of parallel sections in vector bundles

JOURNAL OF GEOMETRY AND PHYSICS

卷:99

Article

Kirschner, Tim¹ 

[1] Univ Duisburg Essen, Fak Math, Essen, Germany

关键词: Linear connections;    Parallel sections;    Extendability;   

DOI  :  10.1016/j.geomphys.2015.10.007

来源: Elsevier

【 摘 要 】

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 del on E, any del-parallel section in E defined on U extends to a del-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. (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

Free