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An intrinsic flat limit of Riemannian manifolds with no geodesics


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.

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Correspondence to C. Sormani.

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J. Basilio was partially supported by NSF DMS 1006059, D. Kazaras by NSF DMS 1547145, and C. Sormani by NSF DMS 1612049.

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Basilio, J., Kazaras, D. & Sormani, C. An intrinsic flat limit of Riemannian manifolds with no geodesics. Geom Dedicata 204, 265–284 (2020).

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Mathematics Subject Classification

  • 53C23