A weakly second order differential structure on rectifiable metric measure spaces

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.

29 pages. Several statements added. The title changed

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• Lorentz meets Lipschitz
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• A new notion of angle between three points in a metric space
• Angles between curves in metric measure spaces
• Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• Spectral convergence under bounded Ricci curvature
• Ricci measure for some singular Riemannian metrics