Hebrew University topology and geometry seminar

Abstract: In this talk I will discuss a new notion of convergence of Weitzenböck manifolds. A Weitzenböck manifold is a Riemannian manifold endowed with a flat metric connection, i.e. it is a triplet (M,g,T), with T being the torsion field of the connection. In material science, these manifolds model bodies with a specific kind of defects called "edge dislocations", which are in a sense "curvature dipoles". This notion of convergence arises when one wants to obtain a body with a continuous distribution of such defects as a limit of manifolds with finitely many ones.
In this talk I will show that a sequence of locally-flat manifolds with an increasing number of defects (or "holes") and no torsion can converge to a manifold with non-zero torsion, and that essentially any compact 2-dimensional Weitzenböck manifold with corners can be obtained in this way. To do so, an extension to non-symmetric connections of some of the classical properties of geodesics is needed. Finally, relations to other notions of convergence will be presented, as well as some open problems.
Based on a joint work with Raz Kupferman. No knowledge in material science will be assumed.