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

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

Keywords

piecewise Euclidean structure, CAT(0) space, Hadamard space, rigidity theorem

Mathematical Subject Classification

Primary: 57S30

Secondary: 53C20

References

Forward citations

Publication

Received: 19 December 1998

Accepted: 27 August 1999

Published: 13 September 1999

Proposed: Steve Ferry

Seconded: Walter Neumann, David Gabai

Authors

Michael W Davis
Department of Mathematics The Ohio State University Columbus Ohio 43201 USA

Boris Okun
Department of Mathematics Vanderbilt University Nashville Tennessee 37400 USA

Fangyang Zheng
Department of Mathematics The Ohio State University Columbus Ohio 43201 USA

Volume 3, issue 1 (1999)

Michael W Davis, Boris Okun and Fangyang Zheng