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Keywords:
weakly $\phi $-symmetric; weakly $\phi $-Ricci symmetric; Kenmotsu manifold; Einstein manifold; $\eta $-Einstein manifold
Summary:
The object of the present paper is to study weakly $\phi $-symmetric and weakly $\phi $-Ricci symmetric Kenmotsu manifolds. It is shown that weakly $\phi $-symmetric and weakly $\phi $-Ricci symmetric Kenmotsu manifolds are $\eta $-Einstein.
References:
[1] Blair, D. E.: Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics 509, Springer, Berlin, 1976. MR 0467588 | Zbl 0319.53026
[2] Cartan, E.: Sur une classe remarquable d’espaces de Riemannian. Bull. Soc. Math. France 54 (1926), 214–264. MR 1504900
[3] Chaki, M. C.: On pseudo-symmetric manifolds. An. Sti. Ale Univ., “AL. I. CUZA" Din Iasi 33 (1987), 53–58. MR 0925690 | Zbl 0634.53012
[4] Chaki, M. C.: On generalized pseudo-symmetric manifolds. Publ. Math. Debrecen 45 (1994), 305–312.
[5] De, U. C.: On $\phi $-symmetric Kenmotsu manifolds. Int. Electronic J. Geom. 1, 1 (2008), 33–38. MR 2390388 | Zbl 1138.53029
[6] De, U. C., Bandyopadhyay, S.: On weakly symmetric Riemannian spaces. Publ. Math. Debrecen 54 (1999), 377–381. MR 1694492 | Zbl 0922.53018
[7] Deszcz R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Ser. A 44, 1 (1992), 1–34. MR 1315367 | Zbl 0808.53012
[8] Hui, S. K., Matsuyama, Y., Shaikh, A. A.: On decomposable weakly conformally symmetric manifolds. Acta Math. Hungar. 128, 1-2 (2010), 82–95. MR 2665800 | Zbl 1224.53057
[9] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24 (1972), 93–103. DOI 10.2748/tmj/1178241594 | MR 0319102 | Zbl 0245.53040
[10] Mikeš, J.: Projective-symmetric and projective-recurrent affinely connected spaces. Tr. Geom. Semin. 13 (1981), 61–62 (in Russian).
[11] Mikeš, J.: Geodesic mappings of special Riemannian spaces. In: Topics in differential geometry, Pap. Colloq., Hajduszoboszló, Hung., 1984, Vol. 2 Colloq. Math. Soc. János Bolyai 46 (1988), 793–813. MR 0933875
[12] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. 78, 3 (1996), 311–333. DOI 10.1007/BF02365193 | MR 1384327
[13] Mikeš, J., Tolobaev, O. S.: Symmetric and projectively symmetric affinely connected spaces. Studies on topological and generalized spaces, Collect. Sci. Works, Frunze (1988), 58–63 (in Russian). MR 1165335
[14] Oubiña, J. A.: New classes of almost contact metric structures. Publ. Math. Debrecen 32 (1985), 187–193. MR 0834769 | Zbl 0611.53032
[15] Özgür, C.: On weakly symmetric Kenmotsu manifolds. Diff. Geom.-Dynamical Systems 8 (2006), 204–209. MR 2220726 | Zbl 1156.53309
[16] Prvanović, M.: On weakly symmetric Riemmanian manifolds. Publ. Math. Debrecen 46 (1995), 19–25.
[17] Roter, W.: On conformally symmetric Ricci recurrent space. Colloq. Math. 31 (1974), 87–96. MR 0372768
[18] Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), 47–87. MR 0088511 | Zbl 0072.08201
[19] Shaikh, A. A., Hui, S. K.: On weakly conharmonically symmetric manifolds. Tensor, N. S. 70 (2008), 119–134. MR 2546909 | Zbl 1193.53115
[20] Shaikh, A. A., Hui, S. K.: On decomposable weakly conharmonically symmetric manifolds. Lobachevski J. Math. 29, 4 (2008), 206–215. DOI 10.1134/S1995080208040021 | MR 2461622 | Zbl 1167.53305
[21] Shaikh, A. A., Hui, S. K.: On weakly concircular symmetric manifolds. Ann. Sti. Ale Univ., “Al. I. CUZA", Din Iasi 55, 1 (2009), 167–186. MR 2510720 | Zbl 1199.53057
[22] Shaikh, A. A., Hui, S. K.: On weakly projective symmetric manifolds. Acta Math. Academiae Paedagogicae Nyiregyhaziensis 25, 2 (2009), 247–269. MR 2570946 | Zbl 1224.53039
[23] Shaikh, A. A., Hui, S. K.: On weak symmetries of trans-Sasakian manifolds. Proc. Estonian Acad. Sci. 58, 4 (2009), 213–223. MR 2604249 | Zbl 1185.53032
[24] Shaikh, A. A., Jana, S. K.: On weakly symmetric Riemannian manifolds. Publ. Math. Debrecen. 71 (2007), 27–41. MR 2340032 | Zbl 1136.53019
[25] Shaikh, A. A., Jana, S. K.: On weakly quasi-conformally symmetric manifolds. SUT. J. Math. 43, 1 (2007), 61–83. MR 2417157 | Zbl 1139.53008
[26] Shukla, S. S., Shukla, M. K.: On $\phi $-Ricci symmetric Kenmotsu manifolds. Novi Sad J. Math. 39, 2 (2009), 89–95. MR 2656183 | Zbl 1224.53063
[27] Sinyukov, N. S.: Geodesic mappings of Riemannian spaces. Nauka, Moscow, 1979, (in Russian). MR 0552022 | Zbl 0637.53020
[28] Szabó, Z. I.: Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R = 0$, The local version. J. Diff. Geom. 17 (1982), 531–582. MR 0683165
[29] Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J. 21 (1969), 21–38. DOI 10.2748/tmj/1178243031 | MR 0242094 | Zbl 1168.51302
[30] Takahashi, T.: Sasakian $\phi $-symmetric spaces. Tohoku Math. J. 29 (1977), 91–113. DOI 10.2748/tmj/1178240699 | MR 0440472
[31] Tamássy, L., Binh, T. Q.: On weakly symmetric and weakly projective symmetric Riemannian manifolds. Coll. Math. Soc. J. Bolyai 56 (1989), 663–670. MR 1211691
[32] Tamássy, L., Binh, T. Q.: On weak symmetrics of Einstein and Sasakian manifolds. Tensor, N. S. 53 (1993), 140–148. MR 1455411
[33] Walker, A. G.: On Ruses spaces of recurrent curvature. Proc. London Math. Soc. 52 (1950), 36–64. MR 0037574
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