Previous |  Up |  Next

Article

Keywords:
pseudo-conformally symmetric manifold; almost pseudo-conformally symmetric manifold; Ricci-recurrent manifold; Einstein field equations; Segre' characteristic
Summary:
The object of the present paper is to study almost pseudo-conformally symmetric Ricci-recurrent manifolds. The existence of almost pseudo-conformally symmetric Ricci-recurrent manifolds has been proved by an explicit example. Some geometric properties have been studied. Among others we prove that in such a manifold the vector field $\rho $ corresponding to the 1-form of recurrence is irrotational and the integral curves of the vector field $\rho $ are geodesic. We also study some global properties of such a manifold. Finally, we study almost pseudo-conformally symmetric Ricci-recurrent spacetime. We obtain the Segre' characteristic of such a spacetime.
References:
[1] Adati, T., Miyazawa, T.: On a Riemannian space with recurrent conformal curvature. Tensor, N. S. 18 (1967), 348-354. MR 0215251 | Zbl 0152.39103
[2] Boeckx, E., Vanhecke, L., Kowalski, O.: Riemannian Manifolds of Conullity Two. World Scientific Publishing Singapore (1996). MR 1462887 | Zbl 0904.53006
[3] Buchner, K., Roter, W.: On conformally quasi-recurrent metrics I. Some general results and existence questions. Soochow J. Math. 19 (1993), 381-400. MR 1249054 | Zbl 0804.53021
[4] Cartan, E.: Sur une classe remarquable d'espaces de Riemannian. Bull. S. M. F 54 (1926), 214-264 France. MR 1504900
[5] Chaki, M. C.: On pseudo symmetric manifolds. Ann. Sţiinţ. Univ. ``Al. I. Cuza'' Iaşi 33 (1987), 53-58. MR 0925690 | Zbl 0634.53012
[6] Chaki, M. C., Gupta, B.: On conformally symmetric spaces. Indian J. Math. 5 (1963), 113-122. MR 0163255 | Zbl 0122.39902
[7] De, U. C., Biswas, H. A.: On pseudo conformally symmetric manifolds. Bull. Calcutta Math. Soc. 85 (1993), 479-486. MR 1326445 | Zbl 0821.53018
[8] De, U. C., Gazi, A. K.: On almost pseudo symmetric manifolds. Ann. Univ. Sci. Budap. 51 (2008), 53-68. MR 2567494 | Zbl 1224.53056
[9] De, U. C., Gazi, A. K.: On almost pseudo conformally symmetric manifolds. Demonstr. Math. 42 (2009), 869-886. MR 2588986 | Zbl 1184.53034
[10] De, U. C., Bandyopadhyay, S.: On weakly conformally symmetric spaces. Publ. Math. 57 (2000), 71-78. MR 1771672 | Zbl 0958.53016
[11] Derdzinski, A., Roter, W.: On compact manifolds admitting indefinite metrics with parallel Weyl tensor. J. Geom. Phys. 58 (2008), 1137-1147. DOI 10.1016/j.geomphys.2008.03.011 | MR 2451274 | Zbl 1154.53049
[12] Derdzinski, A., Roter, W.: Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann. Global Anal. Geom. 37 (2010), 73-90. DOI 10.1007/s10455-009-9173-9 | MR 2575471 | Zbl 1193.53147
[13] Derdzinski, A., Roter, W.: Global properties of indefinite metrics with parallel Weyl tensor. Proc. Pure and Applied Differential Geometry, PADGE 2007 Shaker Aachen (2007), 63-72. MR 2497674 | Zbl 1140.53034
[14] Derdzinski, A., Roter, W.: Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math. J. 59 (2007), 566-602. DOI 10.2748/tmj/1199649875 | MR 2404206 | Zbl 1146.53014
[15] Derdzinski, A., Roter, W.: The local structure of conformally symmetric manifolds. Bull. Belg. Math. Soc. -- Simon Stevin 16 (2009), 117-128. MR 2498963 | Zbl 1165.53011
[16] Deszcz, R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Soc. Sér. A 44 (1992), 1-34. MR 1315367 | Zbl 0808.53012
[17] Eisenhart, L. P.: Riemannian Geometry. Princeton University Press Princeton (1967). MR 0035081 | Zbl 0174.53303
[18] O'Neill, B.: Semi-Riemannian Geometry. With Applications to Relativity. Academic Press New York-London (1983). MR 0719023 | Zbl 0531.53051
[19] Patterson, E. M.: Some theorems on Ricci-recurrent spaces. J. Lond. Math. Soc. 27 (1952), 287-295. DOI 10.1112/jlms/s1-27.3.287 | MR 0048891 | Zbl 0048.15604
[20] Petersen, P.: Riemannian Geometry. Springer New York (2006). MR 2243772 | Zbl 1220.53002
[21] Petrov, A. Z.: Einstein Spaces. Pergamon Press Oxford (1969). MR 0244912 | Zbl 0174.28305
[22] Prvanović, M.: Some theorems on conformally quasi-recurrent manifolds. Fak. Univ. u Novom Sadu, Zb. Rad. Prir.-Mat., Ser. Mat. 19 (1989), 21-31. MR 1099989
[23] Roter, W.: On conformally symmetric Ricci-recurrent spaces. Colloq. Math. 31 (1974), 87-96. MR 0372768 | Zbl 0292.53014
[24] Roter, W.: On the existence of conformally recurrent Ricci-recurrent spaces. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 24 (1976), 973-979. MR 0433341
[25] Roter, W.: Some remarks on infinitesimal projective transformations in recurrent and Ricci-recurrent spaces. Colloq. Math. 15 (1966), 121-127. MR 0211359 | Zbl 0163.43403
[26] Schouten, J. A.: Ricci Calculus. An Introduction to Tensor Analysis and Its Geometrical Application. 2nd ed. Springer Berlin (1954). MR 0066025
[27] Sen, R. N., Chaki, M. C.: On curvature restrictions of a certain kind of conformally-flat Riemannian space of class one. Proc. Natl. Inst. Sci. India, Part A 33 (1967), 100-102. MR 0232308 | Zbl 0163.43401
[28] Suh, Y. J., Kwon, J.-H., Hae, Y. Y.: Conformally symmetric semi-Riemannian manifolds. J. Geom. Phys. 56 (2006), 875-901. DOI 10.1016/j.geomphys.2005.05.005 | MR 2217494
[29] Szabo, Z. I.: Structure theorems on Riemannian spaces satisfying $R(X,Y).R=0$. The local version. J. Differ. Geom. 17 (1982), 531-582. MR 0683165
[30] Tamássy, L., Binh, T. Q.: On weakly symmetric and weakly projectively symmetric Riemannian manifolds. Colloq. János Bolyai Math. Soc. 56 (1989), 663-670. MR 1211691
[31] Walker, A. G.: On Ruse's space of recurrent curvature. Proc. London Math. Soc., II. Sér. 52 (1951), 36-64.
[32] Watanabe, Y.: Integral inequalities in a compact orientable manifold, Riemannian or Kählerian. Kōdai Math. Semin. Rep. 20 (1968), 264-271. DOI 10.2996/kmj/1138845694 | MR 0248702 | Zbl 0172.23302
[33] Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker New York (1970). MR 0284950 | Zbl 0213.23801
Partner of
EuDML logo