Oct 13, 2024
Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in 3-space force uniqueness of geodesics joining two points—implying that any segment of geodesic is also a shortest path.
But there is a long way toward the modern statement: “on any complete abstract Riemannian manifold of ≥0 curvature of any dimension, curvature is the quotient of its universal covering by a discrete group of isometries.”
Marcel Berger, Riemannian Geometry during the Second Half of the Twentieth Century
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Hadamard, 1898 being Les surfaces à courbure opposées et leurs lignes géodésiques
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