On nonpositively curved compact Riemannian manifolds with degenerate Ricci tensor
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- by Dincer Guler and Fangyang Zheng PDF
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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 $\textbf {R}^{n\! -\! r} \! \times \! N^r_a$, 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
- MR Author ID: 272367
- Email: zheng@math.ohio-state.edu
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1265-1285
- MSC (2000): Primary 53C20; Secondary 53C12
- DOI: https://doi.org/10.1090/S0002-9947-2010-05316-4
- MathSciNet review: 2737265