Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 16 (2021), 371 -- 383

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ON GENERALIZED G-RECURRENT MANIFOLDS

Jaeman Kim

Abstract. In this paper, we define a type of Riemannian manifold called generalized G-recurrent manifold, and study the various properties of such a manifold. Among others, it is shown that if a generalized G-recurrent manifold is Einstein, then its associated 1-forms are closed and that if a generalized G-recurrent manifold with constant scalar curvature is conformally flat, then the manifold is semisymmetric. Furthermore a sufficient condition for a generalized G-recurrent manifold to be quasi Einstein is obtained.

2020 Mathematics Subject Classification: 53C15; 53C25
Keywords: generalized G-recurrent manifold; Einstein; conformally flat; semisymmetry; quasi Einstein

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References

  1. K. Arslan, U.C. De, C. Murathan, A. Yildiz: On generalized recurrent Riemannian manifolds, Acta Math.Hungar. 123 (2009), 27-39. MR2496479. Zbl 1199.53166.

  2. A.L. Besse: Einstein Manifolds, Springer, Berlin (1987). MR2371700. Zbl 0613.53001.

  3. M.C. Chaki and M.L. Ghosh: On quasi Einstein manifolds, Ind.Jour.Math. Vol.42, (2000), 211-220. MR1859058. Zbl 1043.53036.

  4. U.C. De and G.C. Ghosh: On quasi Einstein manifolds, Period.Math.Hungar. Vol.48, (2004), 223-231. MR2077698. Zbl 1059.53030.

  5. U.C. De and N. Guha: On generalized recurrent manifolds, J. National Academy of Math. India, 9 (1991), 85-92.

  6. U.C. De, N. Guha and D. Kamilya: On generalized Ricci recurrent manifolds, Tensor (N.S.), 56, (1995), 312-317. MR1413041. Zbl 0859.53009.

  7. U.C. De and D. Kamilya: On generalized conharmonically recurrent manifolds, Indian J.Math., 36, (1994), 49-54. MR1315894. Zbl 0869.53013.

  8. R.S.D. Dubey: Generalized recurrent spaces, Indian J.Pure Appl.Math. 10 (1979), 1508-1513. MR0551789. Zbl 0422.53007.

  9. L.P. Eisenhart: Riemannian Geometry, Princeton University Press, 1949. MR1487892. Zbl 0174.53303.

  10. Y. Ishii: On conharmonic transformations, Tensor N.S. Vol.7, (1957), 73-80. MR0102837. Zbl 0079.15702.

  11. J. Kim: Some properties of N(k)-quasi Einstein manifolds, Int.J.Pure Appl.Math. Vol.88, (2013), no.3, 351-362. Zbl 1322.53035.

  12. J. Kim: On a type of curvature-like tensor, to prepare for publishing.

  13. S. Mallick, A. De and U.C. De: On generalized Ricci recurrent manifolds with applications to relativity, Proceedings of the National Academy of sciences, India Section A: Physical sciences, 83, (2013), 143-152. MR3065589. Zbl 1325.53061.

  14. Y.B. Maralabhavi and M. Rathnamma: On generalized recurrent manifolds, Indian J.Pure Appl.Math. 30 (1999), 1167-1171.

  15. C. Ozgur: On generalized recurrent contact metric manifolds, Indian J.Math. 50, (2008), 11-19. MR2379132. Zbl 1145.53063.

  16. G.P. Pokhariyal and R.S. Mishra: Curvature tensors and their relativistic significance, Yokohama Math.J. Vol.18, (1970), 105-108. MR0292473.

  17. G.P. Pokhariyal and R.S. Mishra: Curvature tensors and their relativistic significance II, Yokohama Math.J. Vol.19, (1971), no.2, 97-103. MR0426797. Zbl 0229.53026.

  18. G.P. Pokhariyal: Curvature tensors and their relativistic significance III, Yokohama Math.J. Vol.21, (1973), 115-119. MR0465066. Zbl 0291.53011.

  19. G.P. Pokhariyal:Relativistic significance of curvature tensors, Int.J.Math.Math.Sci. Vol.5, (1982), no 1, 133-139. MR0666500. Zbl 0486.53022.

  20. B. Prasad: A pseudo projective curvature tensor on a Riemannian manifold, Bull.Calcutta Math.Soc. Vol.94, (2002), no 3, 163-166. MR1859058. Zbl 1028.53016.

  21. J.A. Schouten: Ricci-Calculus. An introduction to tensor analysis and its geometrical applications, Springer-Verlag, Berlin-Gottingen-Heidelberg, (1954). MR0066025. Zbl 0057.37803.

  22. A.A. Shaikh: On pseudo quasi Einstein manifolds, Period.Math.Hungar. Vol.59, (2009), 119-146. MR2587857. Zbl 1224.53029.

  23. H. Singh and Q. Khan: On generalized recurrent Riemannian manifolds, Publ.Math.Debrecen 56 (2000), 87-95. MR1740495. Zbl 0974.53034.

  24. Z.I. Szabo: Structure theorems on Riemannian spaces satisfying R(X,Y)dot R=0 I, the local version, J. Diff. Geometry 17 (1982), 531-582. MR0683165. Zbl 0508.53025.

  25. Z.I. Szabo: Structure theorems on Riemannian spaces satisfying R(X,Y)dot R=0 II, Global versions, Geom. Dedicata 19 (1985), 65-108. MR0797152. Zbl 0612.53023.

  26. M.M. Tripathi and P. Gupta: \Im-curvature tensor on a semi-Riemannian manifold, J.Adv.Math.Stud. Vol.4, (2011), no 1, 117-129. MR2808047. Zbl 1239.53053.

  27. A.G. Walker: On Ruse's spaces of recurrent curvature, Proc.London Math.Soc. Vol.52, (1950), 36-64. MR0037574. Zbl 0039.17702.

  28. K. Yano: Concircular Geometry I. Concircular transformations, Math.Institute,Tokyo Imperial Univ.Proc. Vol.16,(1940), 195-200. MR0003113.

  29. K. Yano and S. Bochner: Curvature and Betti numbers, Ann.Math.Stud. Vol.32, Princeton University Press, 1953. MR0062505. Zbl 0051.39402.

  30. K. Yano and S. Sawaki: Riemannian manifolds admitting a conformal transformation group, J.Differ.Geom. Vol.2, (1968), 161-184. MR0233314. Zbl 0167.19802.






Jaeman Kim
Department of Mathematics Education, Kangwon National University
Chunchon 200-701, Kangwon Do, Korea
e-mail: jaeman64@kangwon.ac.kr

http://www.utgjiu.ro/math/sma