On generalized M-projectively recurrent manifolds

Chand De, Uday ; Pal, Prajjwal

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, no. 13 (2014).

Résumé

The purpose of the present paper is to study generalized M-projectively recurrent manifolds. Some geometric properties of generalized M projectively recurrent manifolds have been studied under certain curvature conditions. An application of such a manifold in the theory of relativity has also been shown. Finally, we give an example of a generalized M-projectively recurrent manifold.

Bibliographie

[1] T. Adati, T. Miyazawa, On Riemannian space with recurrent conformal curvature, Tensor (N.S.)18 (1967), 348-354. Cited on 77.

[2] K. Arslan, U.C. De, C. Murathan, A. Yildiz, On generalized recurrent Riemannian manifolds, Acta Math. Hungar. 123 (2009), no. 1-2, 27-39. Cited on 78.[WoS]

[3] R.L. Bishop, B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. Cited on 92.[Crossref]

[4] E. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France 54 (1926), 214-264. Cited on 77.

[5] M.C. Chaki, Some theorems on recurrent and Ricci-recurrent spaces, Rend. Sem. Mat. Univ. Padova 26 (1956), 168-176. Cited on 78.

[6] M.C. Chaki, B. Gupta, On conformally symmetric spaces, Indian J. Math. 5 (1963), 113-122. Cited on 77.

[7] M.C. Chaki,On pseudo symmetric manifolds, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat. 33 (1987), no. 1, 53-58. Cited on 77.

[8] S.K. Chaubey, R.H. Ojha, On the m-projective curvature tensor of a Kenmotsu manifold Differ. Geom. Dyn. Syst. 12 (2010), 52-60. Cited on 79.

[9] S.K. Chaubey, On weakly m-projectively symmetric manifolds, Novi Sad J. Math. 42 (2012), no. 1, 67-79. Cited on 79.

[10] U.C. De, A.K. Gazi, On generalized concircularly recurrent manifolds, Studia Sci. Math. Hungar. 46 (2009), no. 2, 287-296. Cited on 79.

[11] U.C. De, N. Guha, On generalised recurrent manifolds, Proc. Math. Soc. 7 (1991), 7-11. Cited on 78.

[12] U.C. De, N. Guha, D. Kamilya, On generalized Ricci-recurrent manifolds, Tensor (N.S.) 56 (1995), no. 3, 312-317. Cited on 78.

[13] U.C. De, S. Mallick, Spacetimes admitting M-projective curvature tensor, Bulg. J. Phys. 39 (2012), no. 4, 331-338. Cited on 79.

[14] L.P. Eisenhart, Riemannian Geometry, 2d printing, Princeton University Press, Princeton, N. J., 1949. Cited on 84.

[15] D. Ghosh, On projective recurrent spaces of second order, Acad. Roy. Belg. Bull. Cl. Sci. (5) 56 (1970) 1093-1099. Cited on 77.

[16] M. Hotlos, On conformally symmetric warped products, Ann. Acad. Pedagog. Crac. Stud. Math. 4 (2004), 75-85. Cited on 93.

[17] G.I. Kruckovic, On semireducible Riemannian spaces (Russian), Dokl. Akad. Nauk SSSR (N.S.) 115 (1957), 862-865. Cited on 92.

[18] A. Lichnerowicz, Courbure, nombres de Betti, et espaces symétriques (French), Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 216-223. Amer. Math. Soc., Providence, R. I., 1952. Cited on 77.

[19] S. Mallick, A. De, U.C. De, On generalized Ricci recurrent manifolds with applications to relativity, Proc. Nat. Acad. Sci. India Sect. A 83 (2013), no. 2, 143-152. Cited on 78.

[20] C.A. Mantica, Y.J. Suh, Conformally symmetric manifolds and quasi conformally recurrent Riemannian manifolds, Balkan J. Geom. Appl. 16 (2011), no. 1, 66-77. Cited on 79.

[21] N. Mazumder, R. Biswas, S. Chakraborty, Cosmological evolution across phantom crossing and the nature of the horizons, Astrophys Space Sci 334 (2011), 183-186. Cited on 96.[WoS]

[22] B. O’Neill, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Cited on 77 and 95.

[23] C. Özgür, On generalized recurrent Kenmotsu manifolds, World Appl. Sci. J.2 (2007), no. 1, 9-33. Cited on 78.

[24] C. Özgür, On generalized recurrent contact metric manifolds, Indian J. Math. 50 (2008), no. 1, 11-19. Cited on 78.

[25] C. Özgür, On generalized recurrent Lorentzian para-Sasakian manifolds, Int. J. Appl. Math. Stat. 13 (2008), No. D08, 92-97. Cited on 78.

[26] E.M. Patterson, Some theorems on Ricci-recurrent spaces, J. London Math. Soc. 27 (1952), 287-295. Cited on 77 and 78.[Crossref]

[27] G.P. Pokhariyal, R.S. Mishra, Curvature tensors and their relativistic significance. II, Yokohama Math. J. 19 (1971), no. 2, 97-103. Cited on 79.

[28] N. Prakash, A note on Ricci-recurrent and recurrent spaces Bull. Calcutta Math. Soc. 54 (1962) 1-7. Cited on 78.

[29] W. Roter, On conformally symmetric Ricci-recurrent spaces Colloq. Math. 31 (1974), 87-96. Cited on 77 and 78.

[30] H.S. Ruse, A classification of K*-spaces, Proc. London Math. Soc. (2) 53 (1951), 212-229. Cited on 78.[Crossref]

[31] R.K. Sachs, H. Wu, General relativity for mathematicians, Graduate Texts in Mathematics, Vol. 48, Springer-Verlag, New York-Heidelberg, 1977. Cited on 95.

[32] J.A. Schouten, Ricci-calculus. An introduction to tensor analysis and its geometrical applications, 2d. ed., Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd X. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. Cited on 86 and 88.

[33] J.P. Singh, On m-projective recurrent Riemannian manifold (English summary), Int. J. Math. Anal. (Ruse) 6 (2012), no. 21-24, 1173-1178. Cited on 79.

[34] R.N. Singh, S.K. Pandey, On the m-projective curvature tensor of generalized Sasakian space forms, to appear in Bull. Math. Anal. Appl. Cited on 79.

[35] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Hertl, Exact solutions of Einstein’s field equations. Second edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003. Cited on 96.

[36] A.G. Walker, On Ruse’s spaces of recurrent curvature, Proc. London Math. Soc. (2) 52 (1950), 36-64. Cited on 77 and 78.[Crossref]

[37] S. Yamaguchi, M. Matsumoto, On Ricci-recurrent spaces, Tensor (N.S.) 19 (1968), 64-68. Cited on 78.

[38] K. Yano, M. Kon, Structures on manifolds, Series in Pure Mathematics 3, World Scientific Publishing Co., Singapore, 1984. Cited on 89