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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On nonpositively curved compact Riemannian manifolds with degenerate Ricci tensor


Authors: Dincer Guler and Fangyang Zheng
Journal: Trans. Amer. Math. Soc. 363 (2011), 1265-1285
MSC (2000): Primary 53C20; Secondary 53C12
Published electronically: October 22, 2010
MathSciNet review: 2737265
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Abstract: In this article, we prove that if the Ricci tensor of a compact nonpositively curved manifold $ M^n$ is nowhere negative definite, then it admits local flat factors. To be more precise, let $ U\subseteq M$ be the open subset where the Ricci tensor has maximum rank $ r$. Then for any connected component $ U_a$ of $ U$, its universal cover $ \widetilde{U_a}$ is isometric to $ {\bf R}^{n - r} \times N^r_a$, where $ N^r_a$ is a nonpositively curved manifold with negative Ricci curvature.

In particular, if $ M^n$ is compact, nonpositively curved without Euclidean de Rham factor, and both the manifold and the metric are real analytic, then its Ricci tensor must be negative definite somewhere.


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Additional Information

Dincer Guler
Affiliation: Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211
Email: dincer@math.missouri.edu

Fangyang Zheng
Affiliation: Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174 – and – Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: zheng@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05316-4
PII: S 0002-9947(2010)05316-4
Keywords: Nonpositive sectional curvature, Euclidean de Rham factor, Ricci tensor, Ricci rank, totally geodesic foliation, conullity operator.
Received by editor(s): September 19, 2008
Published electronically: October 22, 2010
Additional Notes: This research was partially supported by an NSF Grant, the Ohio State University, the IMS of CUHK and the CMS of Zhejiang University.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.