A weakly second-order differential structure on rectifiable metric measure spaces

Abstract

We give a definition of angles on the Gromov–Hausdorff limit space of a sequence of complete $n$–dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a weakly second-order differential structure on these spaces and prove that they admit a unique Levi-Civita connection, allowing us to define the Hessian of a twice differentiable function.