Topology and curvature of metric spaces
Parvaneh Joharinad and Jürgen Jost
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Submission date: 01. Apr. 2019
published in: Advances in mathematics, 356 (2019), art-no. 106813
DOI number (of the published article): 10.1016/j.aim.2019.106813
Keywords and phrases: Curvature inequality, discrete metric space, hyperconvex, hyperbolic, intersection of balls, tripod
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We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Busemann, our concept also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivial Čech homology. This suggests a link of our geometrical method to the topological method of persistent homology employed in topological data analysis. We also investigate the geometry of general tripod spaces.