On nonpositively curved compact Riemannian manifolds with degenerate Ricci tensor
Authors:
Dincer Guler and Fangyang Zheng
Journal:
Trans. Amer. Math. Soc. 363 (2011), 12651285
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 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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 K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J., 25 (1973), 425444. MR 0350671 (50:3163)
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 [CCR]
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 S.S. Chen and P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math., 24 (1980), 73103. MR 550653 (82k:53052)
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 S.S. Chern and N. Kuiper, Some theorems on the isometric imbedding of compact Riemannian manifolds in Euclidean space, Ann. of Math., 56 (1952), 422430. MR 0050962 (14:408e)
 [DOZ]
 M. Davis, B. Okun and F. Zheng, Piecewise Euclidean structures and Eberlein's Rigidity Theorem in the singular case, Geometry and Topology, 3 (1999), 303330. MR 1714914 (2000j:53054)
 [DR]
 M. Dajczer and L. Rodriquez, Complete real Kähler minimal submanifolds, J. Reine Angew. Math., 419 (1991), 18. MR 1116914 (92g:53057)
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 [E4]
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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 432101174 – and – Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email:
zheng@math.ohiostate.edu
DOI:
http://dx.doi.org/10.1090/S000299472010053164
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.
