Beltrami's theorem

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This result is valid in all dimensions

This article describes a result related to the sectional curvature of a Riemannian manifold

Statement

If there exists a geodesic mapping between two Riemannian manifolds, and if one of them has constant sectional curvature, the other one also has constant sectional curvature. Note that the value of the constant may differ for the two Riemannian manifolds.