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

Honda, Shouhei

doi:10.2140/gt.2014.18.633

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

Shouhei Honda

Geometry & Topology 18 (2014) 633–668

DOI: 10.2140/gt.2014.18.633

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.

Keywords

Gromov–Hausdorff convergence, Ricci curvature, geometric measure theory

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 53C21

References

Publication

Received: 5 March 2013

Accepted: 28 September 2013

Published: 20 March 2014

Proposed: Dmitri Burago

Seconded: John Lott, Tobias H Colding

Authors

Shouhei Honda
Faculty of Mathmatics Kyushu University 744, Motooka Nishi-Ku Fukuoka 819-0395 Japan

Volume 18, issue 2 (2014)