A weakly second order differential structure on rectifiable metric measure spaces

TL;DR

This paper establishes a weakly second order differential structure on Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounds, enabling the definition of a Levi-Civita connection and Hessian.

Contribution

It introduces a novel weakly second order differential structure on limit spaces and constructs a unique Levi-Civita connection in this setting.

Findings

Angles are well-defined on the limit spaces.

A unique Levi-Civita connection exists on these spaces.

Hessian of twice differentiable functions can be defined.

Abstract

We give the definition of angles on a 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 there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.