## On almost pseudo-conformally symmetric Ricci-recurrent manifolds with applications to relativity.(English)Zbl 1274.53049

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 is proved by an explicit example. Some geometric properties are 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 spacetimes. We obtain the Segre characteristic of such a spacetime.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53B20 Local Riemannian geometry 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53B15 Other connections
Full Text:

### References:

 [1] T. Adati, T. Miyazawa: On a Riemannian space with recurrent conformal curvature. Tensor, N. S. 18 (1967), 348–354. · Zbl 0152.39103 [2] E. Boeckx, L. Vanhecke, O. Kowalski: Riemannian Manifolds of Conullity Two. World Scientific Publishing, Singapore, 1996. · Zbl 0904.53006 [3] K. Buchner, W. Roter: On conformally quasi-recurrent metrics I. Some general results and existence questions. Soochow J. Math. 19 (1993), 381–400. · Zbl 0804.53021 [4] E. Cartan: Sur une classe remarquable d’espaces de Riemannian. Bull. S. M. F 54 (1926), 214–264. (In France.) · JFM 52.0425.01 [5] M. C. Chaki: On pseudo symmetric manifolds. Ann. Sţiinţ. Univ. ”Al. I. Cuza” Iaşi 33 (1987), 53–58. · Zbl 0626.53037 [6] M. C. Chaki, B. Gupta: On conformally symmetric spaces. Indian J. Math. 5 (1963), 113–122. · Zbl 0122.39902 [7] U. C. De, H. A. Biswas: On pseudo conformally symmetric manifolds. Bull. Calcutta Math. Soc. 85 (1993), 479–486. · Zbl 0821.53018 [8] U. C. De, A. K. Gazi: On almost pseudo symmetric manifolds. Ann. Univ. Sci. Budap. 51 (2008), 53–68. · Zbl 1224.53056 [9] U. C. De, A. K. Gazi: On almost pseudo conformally symmetric manifolds. Demonstr. Math. 42 (2009), 869–886. · Zbl 1184.53034 [10] U. C. De, S. Bandyopadhyay: On weakly conformally symmetric spaces. Publ. Math. 57 (2000), 71–78. · Zbl 0958.53016 [11] A. Derdzinski, W. Roter: On compact manifolds admitting indefinite metrics with parallel Weyl tensor. J. Geom. Phys. 58 (2008), 1137–1147. · Zbl 1154.53049 [12] A. Derdzinski, W. Roter: Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann. Global Anal. Geom. 37 (2010), 73–90. · Zbl 1193.53147 [13] A. Derdzinski, W. Roter: Global properties of indefinite metrics with parallel Weyl tensor. Proc. Pure and Applied Differential Geometry, PADGE 2007. Shaker, Aachen, 2007, pp. 63–72. · Zbl 1140.53034 [14] A. Derdzinski, W. Roter: Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math. J. 59 (2007), 566–602. · Zbl 1146.53014 [15] A. Derdzinski, W. Roter: The local structure of conformally symmetric manifolds. Bull. Belg. Math. Soc. – Simon Stevin 16 (2009), 117–128. · Zbl 1165.53011 [16] R. Deszcz: On pseudosymmetric spaces. Bull. Soc. Math. Belg. Soc. Sér. A 44 (1992), 1–34. · Zbl 0808.53012 [17] L. P. Eisenhart: Riemannian Geometry. Princeton University Press, Princeton, 1967. · Zbl 0174.53303 [18] B. O’Neill: Semi-Riemannian Geometry. With Applications to Relativity. Academic Press, New York-London, 1983. [19] E. M. Patterson: Some theorems on Ricci-recurrent spaces. J. Lond. Math. Soc. 27 (1952), 287–295. · Zbl 0048.15604 [20] P. Petersen: Riemannian Geometry. Springer, New York, 2006. [21] A. Z. Petrov: Einstein Spaces. Pergamon Press, Oxford, 1969. [22] M. Prvanović: Some theorems on conformally quasi-recurrent manifolds. Fak. Univ. u Novom Sadu, Zb. Rad. Prir.-Mat., Ser. Mat. 19 (1989), 21–31. · Zbl 0715.53017 [23] W. Roter: On conformally symmetric Ricci-recurrent spaces. Colloq. Math. 31 (1974), 87–96. · Zbl 0292.53014 [24] W. Roter: On the existence of conformally recurrent Ricci-recurrent spaces. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 24 (1976), 973–979. · Zbl 0338.53007 [25] W. Roter: Some remarks on infinitesimal projective transformations in recurrent and Ricci-recurrent spaces. Colloq. Math. 15 (1966), 121–127. · Zbl 0163.43403 [26] J. A. Schouten: Ricci Calculus. An Introduction to Tensor Analysis and Its Geometrical Application. 2nd ed. Springer, Berlin, 1954. [27] R. N. Sen, M. C. Chaki: 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. · Zbl 0163.43401 [28] Y. J. Suh, J.-H. Kwon, Y.Y. Hae: Conformally symmetric semi-Riemannian manifolds. J. Geom. Phys. 56 (2006), 875–901. · Zbl 1092.53017 [29] Z. I. Szabo: Structure theorems on Riemannian spaces satisfying R(X,Y).R = 0. The local version. J. Differ. Geom. 17 (1982), 531–582. · Zbl 0508.53025 [30] L. Tamássy, T.Q. Binh: On weakly symmetric and weakly projectively symmetric Riemannian manifolds. Colloq. János Bolyai Math. Soc. 56 (1989), 663–670. · Zbl 0791.53021 [31] A. G. Walker: On Ruse’s space of recurrent curvature. Proc. London Math. Soc., II. Sér. 52 (1951), 36–64. · Zbl 0039.17702 [32] Y. Watanabe: Integral inequalities in a compact orientable manifold, Riemannian or Kählerian. Kōdai Math. Semin. Rep. 20 (1968), 264–271. · Zbl 0172.23302 [33] K. Yano: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York, 1970. · Zbl 0213.23801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.