A well-known question is whether any Riemannian manifold M of nonpositive sectional curvature admits a piecewise Euclidean metric that is nonpositively curved. Here "nonpositively curved" is in the sense of Aleksandrov and Gromov—that is, it is defined by comparing small triangles in the space with triangles in the Euclidean plane via the "CAT(0)-inequality". (See [BH] or [G] for the precise definition.) Our purpose in this paper is to describe a simple construction that gives an affirmative answer to the question in the case of constant sectional curvature. The most naive approach to this problem does not work, at least not obviously. Namely, given a hyperbolic manifold, first find a triangulation of it by hyperbolic simplices. Next, replace each hyperbolic simplex by a Euclidean simplex with the same edge lengths. Finally, try to prove that the resulting metric is nonpositively curved.
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