Consider a two-dimensional Riemannian surface, locally flat everywhere but for two cone singularities of the same magnitude but opposite signs.

In this talk I will show how manifolds with many such "curvature dipoles" can converge to a Riemannian manifold, while their Levi-Civita connections converge to a metric non-symmetric connection (i.e. a connection with non-zero torsion tensor). Moreover, we will see that on the the disc or the torus, essentially any Riemannian metric and flat, metric connection can be obtained as such limit. If time permits, I will discuss the applications of these results to materials science and elasticity theory.

Based on a joint work with Raz Kupferman.