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On conformally flat manifolds with constant positive scalar curvature


Author: Giovanni Catino
Journal: Proc. Amer. Math. Soc. 144 (2016), 2627-2634
MSC (2010): Primary 53C20, 53C21
DOI: https://doi.org/10.1090/proc/12925
Published electronically: November 4, 2015
MathSciNet review: 3477081
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Abstract: We classify compact conformally flat $ n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $ \mathbb{S}^{n}$ with the round metric, $ \mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $ \mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with a rotationally symmetric Derdzinski metric.


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  • [1] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379-392. MR 0266084 (42 #993)
  • [2] Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700 (2008k:53084)
  • [3] S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797. MR 0018022 (8,230a)
  • [4] Jean-Pierre Bourguignon, The ``magic'' of Weitzenböck formulas, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 251-271. MR 1205158 (94a:58181)
  • [5] Gilles Carron and Marc Herzlich, Conformally flat manifolds with nonnegative Ricci curvature, Compos. Math. 142 (2006), no. 3, 798-810. MR 2231203 (2007b:53078), https://doi.org/10.1112/S0010437X06002016
  • [6] Giovanni Catino, Carlo Mantegazza, and Lorenzo Mazzieri, A note on Codazzi tensors, Math. Ann. 362 (2015), no. 1-2, 629-638. MR 3343892, https://doi.org/10.1007/s00208-014-1135-2
  • [7] Qing-Ming Cheng, Compact locally conformally flat Riemannian manifolds, Bull. London Math. Soc. 33 (2001), no. 4, 459-465. MR 1832558 (2002g:53045), https://doi.org/10.1017/S0024609301008074
  • [8] Georges de Rham, Sur la reductibilité d'un espace de Riemann, Comment. Math. Helv. 26 (1952), 328-344 (French). MR 0052177 (14,584a)
  • [9] A. Derdzinski, Some remarks on the local structure of Codazzi tensors, Global differential geometry and global analysis (Berlin, 1979), Lect. Notes in Math., vol. 838, Springer-Verlag, Berlin, 1981, pp. 243-299.
  • [10] -, On compact Riemannian manifolds with harmonic curvature, Math. Ann. 259 (1982), 144-152.
  • [11] Dennis DeTurck and Hubert Goldschmidt, Regularity theorems in Riemannian geometry. II. Harmonic curvature and the Weyl tensor, Forum Math. 1 (1989), no. 4, 377-394. MR 1016679 (90i:53053), https://doi.org/10.1515/form.1989.1.377
  • [12] Samuel I. Goldberg, On conformally flat spaces with definite Ricci curvature, Kōdai Math. Sem. Rep. 21 (1969), 226-232. MR 0253235 (40 #6450)
  • [13] Matthew J. Gursky, Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic, Indiana Univ. Math. J. 43 (1994), no. 3, 747-774. MR 1305946 (95j:53056), https://doi.org/10.1512/iumj.1994.43.43033
  • [14] Matthew J. Gursky, Conformal vector fields on four-manifolds with negative scalar curvature, Math. Z. 232 (1999), no. 2, 265-273. MR 1718693 (2000i:53054), https://doi.org/10.1007/s002090050514
  • [15] Emmanuel Hebey and Michel Vaugon, Un théorème de pincement intégral sur la courbure concirculaire en géométrie conforme, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 5, 483-488 (French, with English and French summaries). MR 1209271 (94c:53049)
  • [16] Emmanuel Hebey and Michel Vaugon, Effective $ L_p$ pinching for the concircular curvature, J. Geom. Anal. 6 (1996), no. 4, 531-553 (1997). MR 1601401 (99j:53038), https://doi.org/10.1007/BF02921622
  • [17] Sönke Hiepko and Helmut Reckziegel, Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter, Manuscripta Math. 31 (1980), no. 1-3, 269-283 (German, with English summary). MR 576500 (82k:53079), https://doi.org/10.1007/BF01303277
  • [18] Zejun Hu and Haizhong Li, Scalar curvature, Killing vector fields and harmonic one-forms on compact Riemannian manifolds, Bull. London Math. Soc. 36 (2004), no. 5, 587-598. MR 2070435 (2005m:53047), https://doi.org/10.1112/S0024609304003455
  • [19] Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47-62. MR 806701 (86k:53059)
  • [20] Jerry L. Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988), no. 5, 667-681. MR 948075 (89k:35039), https://doi.org/10.1002/cpa.3160410508
  • [21] N. H. Kuiper, On conformally-flat spaces in the large, Ann. of Math. (2) 50 (1949), 916-924. MR 0031310 (11,133b)
  • [22] Maria Helena Noronha, Some compact conformally flat manifolds with nonnegative scalar curvature, Geom. Dedicata 47 (1993), no. 3, 255-268. MR 1235219 (94f:53068), https://doi.org/10.1007/BF01263660
  • [23] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Some characterizations of space-forms, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1817-1828 (electronic). MR 2272150 (2008a:53036), https://doi.org/10.1090/S0002-9947-06-04091-8
  • [24] Patrick J. Ryan, A note on conformally flat spaces with constant scalar curvature, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress, Vol. 2 (On Differential Topology, Differential Geometry and Applications, Dalhousie Univ., Halifax, N.S., 1971) Canad. Math. Congr., Montreal, Que., 1972, pp. 115-124. MR 0487882 (58 #7478)
  • [25] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479-495. MR 788292 (86i:58137)
  • [26] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47-71. MR 931204 (89c:58139), https://doi.org/10.1007/BF01393992
  • [27] Mariko Tani, On a conformally flat Riemannian space with positive Ricci curvature, Tôhoku Math. J. (2) 19 (1967), 227-231. MR 0220213 (36 #3279)
  • [28] Hong-Wei Xu and En-Tao Zhao, $ L^p$ Ricci curvature pinching theorems for conformally flat Riemannian manifolds, Pacific J. Math. 245 (2010), no. 2, 381-396. MR 2608443 (2011c:53062), https://doi.org/10.2140/pjm.2010.245.381
  • [29] Shun-Hui Zhu, The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature, Pacific J. Math. 163 (1994), no. 1, 189-199. MR 1256184 (95d:53045)

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Additional Information

Giovanni Catino
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Email: giovanni.catino@polimi.it

DOI: https://doi.org/10.1090/proc/12925
Keywords: Conformally flat manifold, rigidity
Received by editor(s): January 6, 2015
Received by editor(s) in revised form: August 6, 2015
Published electronically: November 4, 2015
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society

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