An intrinsic flat limit of Riemannian manifolds with no geodesics

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 $m\ge 3$, the manifolds $M_j^m$ may be chosen to have positive scalar curvature.

21 pages, 3 figures, comments welcome