Tim Kirschner will speak on

Extendability of parallel sections in vector bundles

Abstract: I will address the following conjecture (due to Antonio J. Di Scala and Gianni Manno, early in 2014): Let $M$ be a simply-connected (second-countable, Hausdorff) manifold of class $C^\infty$. Let $E$ be a finite-rank vector bundle over $M$ (say real), $\nabla$ a connection on $E$, $U$ an open, dense and connected subset of $M$, and $\sigma$ a $\nabla$-parallel section in $E$ defined on $U$. Then there exists an extension of $\sigma$ to a $\nabla$-parallel section defined on all of $M$. As a matter of fact, I will prove the quoted conjecture under the additional hypothesis that the complement of $U$ in $M$ is a $C^1$ submanifold of $M$, boundary allowed. Furthermore, I will discuss problems that arise when the assumption that the complement of $U$ in $M$ be a submanifold of $M$ is dropped, thus touching upon the general form of the conjecture.