On generalized weakly symmetric $\alpha$-cosymplectic manifolds

Abstract

This study is concerned with some results on generalized weakly symmetric and generalized weakly Ricci-symmetric $\alpha$-cosymplectic manifolds. We prove the necessary and sufficient conditions for an $\alpha$-cosymplectic manifold to be generalized weakly symmetric and generalized weakly Ricci-symmetric. On the basis of these results, we give one proper example of generalized weakly symmetric $\alpha$-cosymplectic manifolds.

Keywords

• Weakly symmetric manifold
• weakly Ricci-symmetric manifold
• generalized weakly symmetric manifold
• generalized weakly Ricci-symmetric manifold
• $\alpha$-cosymplectic manifold

References

1. [1] N. Aktan, M. Yıldırım and C. Murathan, Almost $f$-cosymplectic manifolds, Mediterr. J. Math., 11 (2), 775-787, 2014.
2. [2] M.A. Akyol, Conformal anti-invariant submersions from cosymplectic manifolds, Hacet. J. Math. Stat., 46 (2), 176-192, 2017.
3. [3] K.K. Baishya and P.R. Chowdhury, On Generalized weakly symmetric Kenmotsu man- ifolds, Bol. Soc. Paran. Mat., 39 (6), 211-222, 2021.
4. [4] S. Beyendi, G. Ayar and N. Aktan, On a type of $\alpha$-cosymplectic manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (1), 852-861, 2019.
5. [5] D.E. Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes Math. 509, Springer-Verlag, Berlin, 1976.
6. [6] E. Cartan, Sur une classes remarquable d’espaces de Riemannian, Bull. Soc. Math. France, 54, 214-264, 1926.
7. [7] M.C. Chaki, On pseudo Ricci symmetric manifolds, Bulg. J. Physics, 15 (6), 526-531, 1988.
8. [8] U.C. De and S. Bandyopadhyay, On weakly symmetric spaces, Acta Math. Hung., 87(3), 205-212, 2000.

9. [9] R.S.D. Dubey, Generalized recurrent spaces, Indian J. Pure Appl. Math., 10 (12), 1508-1513, 1979.
10. [10] Y. Gündüzalp and M.A. Akyol, Conformal slant submersions from cosymplectic man- ifolds, Turk. J. Math., 42 (5), 2672-2689, 2018.
11. [11] S.K. Hui, A.A. Shaikh and I. Roy, On totaly umbilical hypersurfaces of weakly con- harmonically symmetric spaces, Global journal of Pure and Applied Mathematics, 10 (4), 28-31, 2010.
12. [12] S.K. Jana and A.A. Shaikh, On quasi-conformally flat weakly Ricci symmetric man- ifolds, Acta Math. Hung., 115 (3), 197-214, 2007.
13. [13] T.W. Kim and H.K. Pak, Canonical foliations of certain classes of almost contact metric structures, Acta Math, Sinica, Eng. Ser. Aug., 21 (4), 841-846, 2005.
14. [14] F. Özen and S. Altay, On weakly and pseudo symmetric Riemannian spaces, Indian J. Pure Appl. Math., 33 (10), 1477-1488, 2001.
15. [15] H. Öztürk, C. Murathan, N. Aktan and A.T. Vanli, Almost $\alpha$-cosymplectic $f$- manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (NS), 60 (1), 211-226, 2014.
16. [16] M. Prvanovic, On weakly symmetric Riemannian manifolds, Pub. Math. Debrecen, 46, 19-25, 1995.
17. [17] A.A. Shaikh and K.K.Baishya, On weakly quasi-conformally symmetric manifolds, Soochow Journal of Mathematics 31 (4), 581-595, 2005.
18. [18] L. Tamassy and T.Q. Binh, On weakly symmetric and weakly projective symmetric Riemannian manifolds, Coll. Math. Soc., J. Bolyai, 56, 663-670, 1989.
19. [19] M. Tarafdar and M.A.A. Jawarneh, Semi-Pseudo Ricci Symmetric manifold, J. Indian Inst. Sci., 73, 591-596, 1993.
20. [20] A.G. Walker, On Ruses space of recurrent curvature, Proc. London Math. Soc., 52, 36-54, 1950.
21. [21] M. Yıldırım and S. Beyendi, On almost generalized weakly symmetric $\alpha$-cosymplectic manifolds, Univers. J. Math. Appl., 3 (4), 156-159, 2020.