Volume 18, issue 2 (2014)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A weakly second-order differential structure on rectifiable metric measure spaces

Shouhei Honda

Geometry & Topology 18 (2014) 633–668
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