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An intrinsic flat limit of Riemannian manifolds with no geodesics

Research paper by J. Basilio, D. Kazaras, C. Sormani

Indexed on: 29 Oct '19Published on: 13 May '19Published in: Geometriae Dedicata



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.