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In this article, we generalize Eberlein's Rigidity Theorem to the singularcase, namely, one of the spaces is only assumed to be a CAT(0) topologicalmanifold. As a corollary, we get that any compact irreducible but locallyreducible locally symmetric space of noncompact type does not admit anonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure.Any hyperbolic manifold, on the other hand, does admit such a structure.

Piecewise Euclidean structures and Eberlein’s Rigidity Theorem in the singular case