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.

**[A]**Kinetsu Abe,*Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions*, Tôhoku Math. J. (2)**25**(1973), 425–444. MR**0350671****[BGS]**Werner Ballmann, Mikhael Gromov, and Viktor Schroeder,*Manifolds of nonpositive curvature*, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR**823981****[BS]**V. Bangert and V. Schroeder,*Existence of flat tori in analytic manifolds of nonpositive curvature*, Ann. Sci. École Norm. Sup. (4)**24**(1991), no. 5, 605–634. MR**1132759****[CCR]**Jianguo Cao, Jeff Cheeger, and Xiaochun Rong,*Splittings and Cr-structures for manifolds with nonpositive sectional curvature*, Invent. Math.**144**(2001), no. 1, 139–167. MR**1821146**, 10.1007/PL00005800**[CCR1]**Jianguo Cao, Jeff Cheeger, and Xiaochun Rong,*Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3*, Comm. Anal. Geom.**12**(2004), no. 1-2, 389–415. MR**2074883****[CE]**Su Shing Chen and Patrick Eberlein,*Isometry groups of simply connected manifolds of nonpositive curvature*, Illinois J. Math.**24**(1980), no. 1, 73–103. MR**550653****[CK]**Shiing-shen Chern and Nicolaas H. Kuiper,*Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space*, Ann. of Math. (2)**56**(1952), 422–430. MR**0050962****[DOZ]**Michael W. Davis, Boris Okun, and Fangyang Zheng,*Piecewise Euclidean structures and Eberlein’s rigidity theorem in the singular case*, Geom. Topol.**3**(1999), 303–330 (electronic). MR**1714914**, 10.2140/gt.1999.3.303**[DR]**Marcos Dajczer and Lucio Rodríguez,*Complete real Kähler minimal submanifolds*, J. Reine Angew. Math.**419**(1991), 1–8. MR**1116914**, 10.1007/BF01245181**[E1]**Patrick Eberlein,*Lattices in spaces of nonpositive curvature*, Ann. of Math. (2)**111**(1980), no. 3, 435–476. MR**577132**, 10.2307/1971104**[E2]**Patrick Eberlein,*A canonical form for compact nonpositively curved manifolds whose fundamental groups have nontrivial center*, Math. Ann.**260**(1982), no. 1, 23–29. MR**664362**, 10.1007/BF01475751**[E3]**Patrick Eberlein,*Isometry groups of simply connected manifolds of nonpositive curvature. II*, Acta Math.**149**(1982), no. 1-2, 41–69. MR**674166**, 10.1007/BF02392349**[E4]**Patrick Eberlein,*Euclidean de Rham factor of a lattice of nonpositive curvature*, J. Differential Geom.**18**(1983), no. 2, 209–220. MR**710052****[E5]**Patrick Eberlein,*Symmetry diffeomorphism group of a manifold of nonpositive curvature. II*, Indiana Univ. Math. J.**37**(1988), no. 4, 735–752. MR**982828**, 10.1512/iumj.1988.37.37036**[EHS]**Patrick Eberlein, Ursula Hamenstädt, and Viktor Schroeder,*Manifolds of nonpositive curvature*, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 179–227. MR**1216622****[F]**Dirk Ferus,*On the completeness of nullity foliations*, Michigan Math. J.**18**(1971), 61–64. MR**0279733****[Fl]**Luis A. Florit,*On submanifolds with nonpositive extrinsic curvature*, Math. Ann.**298**(1994), no. 1, 187–192. MR**1252825**, 10.1007/BF01459733**[FW]**Gerd Fischer and H. Wu,*Developable complex analytic submanifolds*, Internat. J. Math.**6**(1995), no. 2, 229–272. MR**1316302**, 10.1142/S0129167X9500047X**[FZ]**Luis A. Florit and Fangyang Zheng,*On nonpositively curved Euclidean submanifolds: splitting results*, Comment. Math. Helv.**74**(1999), no. 1, 53–62. MR**1677106**, 10.1007/s000140050076**[FZ1]**Luis A. Florit and Fangyang Zheng,*On nonpositively curved Euclidean submanifolds: splitting results. II*, J. Reine Angew. Math.**508**(1999), 1–15. MR**1676866**, 10.1515/crll.1999.027**[GZ]**Dincer Guler and Fangyang Zheng,*On Ricci rank of Cartan-Hadamard manifolds*, Internat. J. Math.**13**(2002), no. 6, 557–578. MR**1915520**, 10.1142/S0129167X02001459**[JY]**Jürgen Jost and Shing-Tung Yau,*Harmonic maps and rigidity theorems for spaces of nonpositive curvature*, Comm. Anal. Geom.**7**(1999), no. 4, 681–694. MR**1714940**, 10.4310/CAG.1999.v7.n4.a1**[S1]**Viktor Schroeder,*Existence of immersed tori in manifolds of nonpositive curvature*, J. Reine Angew. Math.**390**(1988), 32–46. MR**953675**, 10.1515/crll.1988.390.32**[S2]**Viktor Schroeder,*Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature*, Manuscripta Math.**64**(1989), no. 1, 77–105. MR**994382**, 10.1007/BF01182086**[S3]**Viktor Schroeder,*Codimension one tori in manifolds of nonpositive curvature*, Geom. Dedicata**33**(1990), no. 3, 251–263. MR**1050413**, 10.1007/BF00181332**[S4]**Viktor Schroeder,*Analytic manifolds of nonpositive curvature with higher rank subspaces*, Arch. Math. (Basel)**56**(1991), no. 1, 81–85. MR**1082005**, 10.1007/BF01190084**[S5]**V. Schroeder, private communications.**[W]**H. Wu,*Complete developable submanifolds in real and complex Euclidean spaces*, Internat. J. Math.**6**(1995), no. 3, 461–489. MR**1327160**, 10.1142/S0129167X95000171**[WZ]**Hung-Hsi Wu and Fangyang Zheng,*On complete developable submanifolds in complex Euclidean spaces*, Comm. Anal. Geom.**10**(2002), no. 3, 611–646. MR**1912259**, 10.4310/CAG.2002.v10.n3.a6**[WZ1]**Hung-Hsi Wu and Fangyang Zheng,*Compact Kähler manifolds with nonpositive bisectional curvature*, J. Differential Geom.**61**(2002), no. 2, 263–287. MR**1972147****[WZ2]**Hung-Hsi Wu and Fangyang Zheng,*Kähler manifolds with slightly positive bisectional curvature*, Explorations in complex and Riemannian geometry, Contemp. Math., vol. 332, Amer. Math. Soc., Providence, RI, 2003, pp. 305–325. MR**2018347**, 10.1090/conm/332/05944**[Z]**Fangyang Zheng,*Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds*, Comment. Math. Helv.**77**(2002), no. 2, 221–234. MR**1915039**, 10.1007/s00014-002-8337-z**[Z1]**Fangyang Zheng,*Ricci degeneracy in non-positively curved manifolds*, Second International Congress of Chinese Mathematicians, New Stud. Adv. Math., vol. 4, Int. Press, Somerville, MA, 2004, pp. 441–450. MR**2498000**

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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

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.