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 | References | Similar Articles | Additional Information

Abstract: In this article, we prove that if the Ricci tensor of a compact nonpositively curved manifold is nowhere negative definite, then it admits local flat factors. To be more precise, let be the open subset where the Ricci tensor has maximum rank . Then for any connected component of , its universal cover is isometric to , where is a nonpositively curved manifold with negative Ricci curvature.

In particular, if 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:
https://doi.org/10.1090/S0002-9947-2010-05316-4

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.