Abstract
Let \(M\) be a \(3\)-dimensional almost contact metric manifold satisfying \((*)\) condition. We denote such a manifold by \(M^{*}\). At first we study symmetric and skew-symmetric parallel tensor of type \((0,2)\) in \(M^{*}\). Next we prove that a non-cosymplectic manifold \(M^{*}\) is Ricci semisymmetric if and only if it is Einstein. Also we study locally \(\phi \)-symmetry and \(\eta \)-parallel Ricci tensor of \(M^{*}\). Finally, we prove that if a non-cosymplectic \(M^{*}\) is Einstein, then the manifold is Sasakian.
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References
Agaoka, Y., Kim, B.H., Choi, J.H.: On 3-dimensional contact metric manifolds. Mem. Fac. Integr. Arts Sci. Hiroshima Univ. 28, 29–33 (2002)
Blair, D..E: Riemannian Geometry Contact and Symplectic Manifolds. Progress in Mathematics in Mathematics, vol. 203. Birkhauser Boston Inc, Boston (2002)
De, U.C., Sarkar, A.: On three-dimensional Trans-Sasakian manifolds. Extracta Math. 23, 265–277 (2008)
De, U.C., Sarkar, A.: Onthree-dimensional quasi-Sasakian manifolds. SUT J. Math. 45, 59–71 (2009)
Gouli-Andreu, F., Xenas, P.J.: On a class of 3-dimensional contact metric manifold. J. Geom. 63, 64–75 (1998)
Jun, J.B., Kim, I.B., Kim, U.K.: On 3-dimensional almost contact metric manifolds. Kyungpook Math. J. 34, 293–301 (1994)
Kon, M.: Invariant submanifolds in Sasakian manifolds. Math. Ann. 219, 227–290 (1976)
Kwon, J.H., Kim, B.H.: A new class of almost contact Riemannian manifolds. Commun. Korean Math. Soc. 8, 455–465 (1993)
Sasaki, S: Almost Contact Manifolds, Lecture Notes, vol. 1. Mathematical Institute, Tohoku University (1965)
Szabo, Z.: Structure theorems on Riemannian spaces satisfying R(X, Y). R = 0, I. The local version. J. Differ. Geom. 17, 531–582 (1982)
Tanno, S.: Promenades on Spheres. Tokyo Institute of Technology, Tokyo (1996)
Takahashi, T.: Sasakian \({\phi }\)-symmetric spaces. Tohoku Math. J. 29, 91–113 (1977)
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De, U.C., Yildiz, A. & Çetinkaya, A. Certain results on a type of contact metric manifold. Afr. Mat. 26, 1229–1236 (2015). https://doi.org/10.1007/s13370-014-0282-7
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DOI: https://doi.org/10.1007/s13370-014-0282-7
Keywords
- Almost contact metric manifold
- Ricci semisymmetric
- Locally \(\phi \)-symmetry
- \(\eta \)-parallel Ricci tensor