Ricci curvature and Orientability

TL;DR

This paper introduces a notion of orientability for measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounds, establishing fundamental properties and relationships with convergence theories.

Contribution

It defines orientability for metric measure spaces, proves stability and uniqueness of orientations, and links convergence of orientations with various geometric convergence notions.

Findings

Orientability is stable under noncollapsed limits.

Only two orientations are possible on a limit space.

Compatibility between different convergence types is established.

Abstract

In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between $L^2$-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and the Gromov-Hausdorff convergence, which is a generalization of a recent work by Matveev-Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.