Because of the attack on the computer network of the University of Duisburg-Essen, some content (in particular some image files) cannot be accessed because it is stored on central servers of the university.

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.