An intrinsic flat limit of Riemannian manifolds with no geodesics

• DOI: 10.1007/s10711-019-00453-1
• Publication Date: 2019-05-14T01:39:17Z
• Authors: J. Basilio; D. Kazaras; C. Sormani
• Keywords: Geodesic map; Riemannian manifold; Riemannian Geometry

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.<br/>21 pages, 3 figures, comments welcome<br/>