An intrinsic flat limit of Riemannian manifolds with no geodesics

TL;DR

This paper constructs a sequence of Riemannian manifolds that converge in the intrinsic flat sense to a sphere lacking geodesics, revealing new behaviors of intrinsic flat limits distinct from Gromov-Hausdorff limits.

Contribution

It demonstrates the existence of intrinsic flat limits of Riemannian manifolds that are not geodesic spaces, contrasting with Gromov-Hausdorff limits, and shows these can have positive scalar curvature.

Findings

Intrinsic flat limits can lack geodesics between points.

Gromov-Hausdorff limits of Riemannian manifolds are always geodesic.

Constructed manifolds can have positive scalar curvature for dimensions ≥ 3.

Abstract

In this paper we produce a sequence of Riemannian manifolds $M_j^m$, $m \ge 2$, which converge in the intrinsic flat sense to the unit $m$-sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat limits. In contrast, any compact Gromov-Hausdorff limit of a sequence of Riemannian manifolds is a geodesic space. Moreover, if $m\geq3$, the manifolds $M_j^m$ may be chosen to have positive scalar curvature.