Abstract
It is shown that a conformally flat contact metric 3-manifold with Ricci curvature vanishing along the characteristic vector field, has non-positive scalar curvature. Such a manifold is flat if (i) it is compact, or (ii) the scalar curvature is constant, or (iii) the norm of the Ricci tensor is constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Bang, Riemannian geometry of vector bundles, Thesis, Michigan State University, 1994.
D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer Verlag, Berlin, 1976.
D.E. Blair, On the non-existence of flat contact metric structures, Tohoku Math. J. 28(1976), 373–379.
D.E. Blair and T. Koufogiorgos, When is the tangent sphere bundle conformally flat? J. Geom. 49(1994), 55–66.
D.E. Blair and R. Sharma, Three dimensional locally symmetric contact metric manifolds, Boll. Un. Mat. Ital. (7), 4-A(1990), 385–390.
J.P. Bourguignon, Harmonic curvature for gravitational and Yang-Mills fields, Lecture Notes in Math., 949, Springer Verlag, Berlin, 1982.
G. Calvaruso, D. Perrone and L. Vanhecke, Homogeneity on three-dimensional contact metric manifolds, Israel J. Math. 114 (1999), 301–321.
Q. Cheng, S. Ishikawa and K. Shiohama, Conformally flat 3-manifolds with constant scalar curvature, J. Math. Soc. Japan 51(1999), 209–226.
L.P. Eisenhart, Symmetric tensors of the second order whose covariant derivatives vanish, Annals of Maths. 27(1926), 91–98.
A. Ghosh, T. Koufogiorgos and R. Sharma, Conformally flat contact metric manifolds, J. Geom. 70(2001), 66–76.
F. Gouli-Andreou and P.J. Xenos, Two classes of conformally flat contact metric manifolds, J. Geom. 64(1999), 80–88.
M. Okumura, Some remarks on spaces with certain contact structures, Tohoku Math. J. 14(1962), 135–145.
P. Rukimbira, A characterization of flat contact metric geometry, Houston J. Math. 24(1998), 409–414.
S. Tanno, Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43(1967), 581–583.
Author information
Authors and Affiliations
Additional information
Dedicated to the memory of Professor S. Tanno
Rights and permissions
About this article
Cite this article
Gouli-Andreou, F., Sharma, R. A Class of Conformally flat Contact Metric 3-Manifolds. Results. Math. 43, 114–120 (2003). https://doi.org/10.1007/BF03322727
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322727