Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
weakly $W_3$-symmetric manifold; $W_3$-curvature tensor; decomposable manifold; scalar curvature; totally umbilical hypersurfaces; totally geodesic; mean curvature
weakly $W_3$-symmetric manifold; $W_3$-curvature tensor; decomposable manifold; scalar curvature; totally umbilical hypersurfaces; totally geodesic; mean curvature
Summary:
The object of the present paper is to study weakly $W_3$-symmetric manifolds and its decomposability with the existence of such notions. Among others it is shown that in a decomposable weakly $W_3$-symmetric manifold both the decompositions are weakly Ricci symmetric.
References:
[1] Altay, S.: Some applications on weakly pseudosymmetric Riemannian manifolds. Diff. Geom. Dyn. Syst. 7 (2005), 1–10. MR 2141667
[2] Binh, T. Q.: On weakly symmetric Riemannian spaces. Publ. Math. Debrecen 42 (1993), 103–107. MR 1208855 | Zbl 0797.53041
[3] Cartan, E.: Sur une classe remarquable d’espaces de Riemannian. Bull. Soc. Math. France 54 (1926), 214–264. MR 1504900
[4] 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
[5] Chaki, M. C.: On generalized pseudo-symmetric manifolds. Publ. Math. Debrecen 45 (1994), 305–312.
[6] Chen, B. Y.: Geometry of submanifolds. Marcel-Deker, New York, 1973. MR 0353212 | Zbl 0262.53036
[7] De, U. C., Bandyopadhyay, S.: On weakly symmetric Riemannian spaces. Publ. Math. Debrecen 54 (1999), 377–381. MR 1694492 | Zbl 0922.53018
[8] Deszcz R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Ser. A 44, 1 (1992), 1–34. MR 1315367 | Zbl 0808.53012
[9] Eisenhart, L. P.: Riemannian Geometry. Princeton University Press, Princeton, 1949. MR 0035081 | Zbl 0041.29403
[10] 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
[11] Mikeš, J.: Projective-symmetric and projective-recurrent affinely connected spaces. Tr. Geom. Semin. 13 (1981), 61–62 (in Russian).
[12] 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
[13] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. 78, 3 (1996), 311–333. DOI 10.1007/BF02365193 | MR 1384327
[14] 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
[15] Özen, F., Altay, S.: On weakly and pseudo symmetric Riemannian spaces. Indian J. Pure Appl. Math. 33, 10 (2001), 1477–1488. MR 1941070
[16] Özen, F., Altay, S.: On weakly and pseudo concircular symmetric structures on a Riemannian manifold. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 47 (2008), 129–138. MR 2482723 | Zbl 1184.53022
[17] Pokhariyal, G. P.: Curvature tensors and their relativistic significance III. Yokohama Math. J. 21 (1973), 115–119. MR 0465066 | Zbl 0291.53011
[18] Prvanović, M.: On weakly symmetric Riemmanian manifolds. Publ. Math. Debrecen 46 (1995), 19–25.
[19] Roter, W.: On conformally symmetric Ricci recurrent space. Colloq. Math. 31 (1974), 87–96. MR 0372768
[20] Schouten, J. A.: Ricci-Calculus, An introduction to Tensor Analysis and its Geometrical Applications. Springer-Verlag, Berlin, 1954. MR 0066025 | Zbl 0057.37803
[21] 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
[22] Shaikh, A. A., Baishya, K. K.: On weakly quasi-conformally symmetric manifolds. Soochow J. Math. 31, 4 (2005), 581–595. MR 2190202 | Zbl 1091.53017
[23] Shaikh, A. A., Hui, S. K.: On weakly conharmonically symmetric manifolds. Tensor, N. S. 70 (2008), 119–134. MR 2546909 | Zbl 1193.53115
[24] 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
[25] 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
[26] 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
[27] Shaikh, A. A., Jana, S. K.: On weakly symmetric Riemannian manifolds. Publ. Math. Debrecen. 71 (2007), 27–41. MR 2340032 | Zbl 1136.53019
[28] 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
[29] Shaikh, A. A., Jana, S. K, Eyasmin, S.: On weakly pseudo quasi-conformally symmetric manifolds. Indian J. Math. 50, 3 (2008), 505–518. MR 2483698 | Zbl 1167.53023
[30] Shaikh, A. A., Roy, I., Hui, S. K.: On totally umbilical hypersurfaces of weakly conharmonically symmetric spaces. Global J. Science Frontier Research 10, 4 (2010), 28–30.
[31] Shaikh, A. A., Shahid, M. H., Hui, S. K.: On weakly conformally symmetric manifolds. Matematiki Vesnik 60 (2008), 269–284. MR 2465809 | Zbl 1224.53033
[32] Sinyukov, N. S.: Geodesic mappings of Riemannian spaces. Nauka, Moscow, 1979, (in Russian). MR 0552022 | Zbl 0637.53020
[33] 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
[34] 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
[35] Tamássy, L., Binh, T. Q.: On weak symmetrics of Einstein and Sasakian manifolds. Tensor, N. S. 53 (1993), 140–148. MR 1455411
[36] Walker, A. G.: On Ruses spaces of recurrent curvature. Proc. London Math. Soc. 52 (1950), 36–64. MR 0037574
[37] Yano, K., Kon, M.: Structure on manifolds. World Scientific Publ., Singapore, 1984. MR 0794310
Search
Partner of
