Classifications of N(k)-contact metric manifolds satisfying certain curvature conditions
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Abstract
The object of the present paper is to classify $N(k)$-contact metric manifolds satisfying certain curvature conditions on the projective curvature tensor. Projectively pseudosymmetric and pseudoprojectively at $N(k)$-contact metric manifolds are considered. Beside these we also study $N(k)$-contact metric manifolds satisfying $\tilde(Z)\dot P = 0$, where \tilde(Z)$ and $P$ denote respectively the concircular and projective curvature tensor. Finally, we give an example of a $N(k)$-contact metric manifold.
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Majhi, P., & De, U.
(2015).
Classifications of N(k)-contact metric manifolds satisfying certain curvature conditions.
Acta Mathematica Universitatis Comenianae, 84(1), 167-178.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/118/134
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References
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at contact metric manifolds, Balkan Journal of Geom. Appl. 5, 2 (2000), 1-7.
[2] D. E. Blair, contact manifolds in Riemannian geometry, Lecture note in Math. 509, Springer-Verlag, Berlin-New York 1976.
[3] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhauser Boston, Inc., Boston 2002.
[4] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. 29 (1977), 319-324.
[5] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), 189-214.
[6] D. E. Blair, J. S. Kim and M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42, 5 (2005), 883-992.
[7] E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian manifolds of conullity two, Singapore World Sci. Publishing 1996.
[8] E. Boeckx, A full classication of contact metric (k, µ)-spaces, Illinois J. Math. 44 (2010), 212-219.
[9] E. Cartan, Sur une classe remarqable d' espaces de Riemannian, Bull. Soc. Math. France. 54 (1962), 214-264.
[10] U. C. De, C. Murathan and K. Arsalan, On the Weyl projective curvature tensor of an N(k)-contact metric manifold, Mathematica Panonoica, 21, 1 (2010), 129-142.
[11] U. C. De and A. Sarkar, On the quasi-conformal curvature tensor of a (k, µ)-contact metric manifold, Math. Reports, 14(64), 2 (2012), 115-129.
[12] U. C. De, A. Yildiz and S. Ghosh, On a class of N(k)-contact metric manifolds, Math. Reports., 16(66)(2014), No. 2.
[13] J. B. Jun and U. K. Kim, On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34, 2 (1994), 293-301.
[14] O. Kowalski, An explicit classification of 3- dimensional Riemannian spaces satisfying R(X, Y ).R = 0, Czechoslovak Math. J. 46, 121 (1996), 427-474.
[15] C. Ozgur, Contact metric manifolds with cyclic-parallel Ricci tensor, Diff. Geom. Dynamical systems, 4 (2002), 21-25.
[16] B. J. Papantoniou, contact Riemannian manifolds satisfying R(ξ,X).R = 0 and ξ ∈ (k, µ)-nullity distribution, Yokohama Math. J. 40 (1993), 149-161.
[17] G. Soos, Ueber die geodaetischen Abbildungen von Riemannaschen Raeumen auf projektiv symmetrische Riemannsche Raeume, Acta. Math. Acad.Sci. Hungar. ica 9 (1958), 359-361.
[18] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y ).R = 0, the local version, J. Diff. Geom. 17 (1982), 531-582.
19] S. Tanno, Ricci curvature of contact metric Riemannian manifolds, Tohoku Math. J. 40 (1988), 441-448.
[20] L. Verstraelen, Comments on pseudosymmetry in the sense of Ryszard Deszcz, In: Geometry and Topology of submanifolds, VI. River Edge, NJ: World Sci. Publishing, 1994, 199-209.
[21] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of mathematics studies, 32, Princeton university press, 1953.
[22] K. Yano, Concircular geometry I. concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200.
at contact metric manifolds, Balkan Journal of Geom. Appl. 5, 2 (2000), 1-7.
[2] D. E. Blair, contact manifolds in Riemannian geometry, Lecture note in Math. 509, Springer-Verlag, Berlin-New York 1976.
[3] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhauser Boston, Inc., Boston 2002.
[4] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. 29 (1977), 319-324.
[5] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), 189-214.
[6] D. E. Blair, J. S. Kim and M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42, 5 (2005), 883-992.
[7] E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian manifolds of conullity two, Singapore World Sci. Publishing 1996.
[8] E. Boeckx, A full classication of contact metric (k, µ)-spaces, Illinois J. Math. 44 (2010), 212-219.
[9] E. Cartan, Sur une classe remarqable d' espaces de Riemannian, Bull. Soc. Math. France. 54 (1962), 214-264.
[10] U. C. De, C. Murathan and K. Arsalan, On the Weyl projective curvature tensor of an N(k)-contact metric manifold, Mathematica Panonoica, 21, 1 (2010), 129-142.
[11] U. C. De and A. Sarkar, On the quasi-conformal curvature tensor of a (k, µ)-contact metric manifold, Math. Reports, 14(64), 2 (2012), 115-129.
[12] U. C. De, A. Yildiz and S. Ghosh, On a class of N(k)-contact metric manifolds, Math. Reports., 16(66)(2014), No. 2.
[13] J. B. Jun and U. K. Kim, On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34, 2 (1994), 293-301.
[14] O. Kowalski, An explicit classification of 3- dimensional Riemannian spaces satisfying R(X, Y ).R = 0, Czechoslovak Math. J. 46, 121 (1996), 427-474.
[15] C. Ozgur, Contact metric manifolds with cyclic-parallel Ricci tensor, Diff. Geom. Dynamical systems, 4 (2002), 21-25.
[16] B. J. Papantoniou, contact Riemannian manifolds satisfying R(ξ,X).R = 0 and ξ ∈ (k, µ)-nullity distribution, Yokohama Math. J. 40 (1993), 149-161.
[17] G. Soos, Ueber die geodaetischen Abbildungen von Riemannaschen Raeumen auf projektiv symmetrische Riemannsche Raeume, Acta. Math. Acad.Sci. Hungar. ica 9 (1958), 359-361.
[18] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y ).R = 0, the local version, J. Diff. Geom. 17 (1982), 531-582.
19] S. Tanno, Ricci curvature of contact metric Riemannian manifolds, Tohoku Math. J. 40 (1988), 441-448.
[20] L. Verstraelen, Comments on pseudosymmetry in the sense of Ryszard Deszcz, In: Geometry and Topology of submanifolds, VI. River Edge, NJ: World Sci. Publishing, 1994, 199-209.
[21] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of mathematics studies, 32, Princeton university press, 1953.
[22] K. Yano, Concircular geometry I. concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200.