Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RIEMANNIAN MANIFOLDS WITH A SEMI-SYMMETRIC METRIC P-CONNECTION

  • Chaubey, Sudhakar Kr (Section of Mathematics Department of Information Technology Shinas College of Technology) ;
  • Lee, Jae Won (Department of Mathematics Education and RINS Gyeongsang National University) ;
  • Yadav, Sunil Kr (Department of Mathematics Poornima College of Engineering)
  • Received : 2018.09.21
  • Accepted : 2019.03.04
  • Published : 2019.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

We define a class of semi-symmetric metric connection on a Riemannian manifold for which the conformal, the projective, the concircular, the quasi conformal and the m-projective curvature tensors are invariant. We also study the properties of semisymmetric, Ricci semisymmetric and Eisenhart problems for solving second order parallel symmetric and skew-symmetric tensors on the Riemannian manifolds equipped with a semi-symmetric metric P-connection.

Keywords

References

  1. B. Barua and A. Kr. Ray, Some properties of semisymmetric metric connection in a Riemannian manifold, Indian J. Pure Appl. Math. 16 (1985), no. 7, 736-740.
  2. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 361-368.
  3. S. K. Chaubey, Certain results on N(k)-quasi Einstein manifolds, Afr. Mat. (2018); https://doi.org/10.1007/s13370-018-0631-z.
  4. S. K. Chaubey and A. Kumar, Semi-symmetric metric T-connection in an almost contact metric manifold, Int. Math. Forum 5 (2010), no. 21-24, 1121-1129.
  5. S. K. Chaubey and R. H. Ojha, On the m-projective curvature tensor of a Kenmotsu manifold, Differ. Geom. Dyn. Syst. 12 (2010), 52-60.
  6. S. K. Chaubey and R. H. Ojha, On a semi-symmetric non-metric connection, Filomat 26 (2012), no. 2, 269- 275. https://doi.org/10.2298/FIL1202269C
  7. S. K. Chaubey and A. A. Shaikh, On 3-dimensional Lorentzian concircular structure manifolds, Commun. Korean Math. Soc. 34 (2019), no. 1, 303-319. https://doi.org/10.4134/CKMS.C180044
  8. S. K. Chaubey and S. K. Yadav, Study of Kenmotsu manifolds with semi-symmetric metric connection, Universal J. Math. Appl. 1 (2018), no. 2, 89-97.
  9. M. Crasmareanu, Parallel tensors and Ricci solitons in N(k)-quasi Einstein manifolds, Indian J. Pure Appl. Math. 43 (2012), no. 4, 359-369. https://doi.org/10.1007/s13226-012-0022-3
  10. U. C. De and J. Sengupta, On a type of semi-symmetric metric connection on an almost contact metric manifold, Facta Univ. Ser. Math. Inform. 16 (2001), 87-96.
  11. L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc. 25 (1923), no. 2, 297-306. https://doi.org/10.1090/S0002-9947-1923-1501245-6
  12. A. Friedmann and J. A. Schouten, Uber die Geometrie der halbsymmetrischen Ubertragungen, Math. Z. 21 (1924), no. 1, 211-223. https://doi.org/10.1007/BF01187468
  13. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
  14. H. A. Hayden, Sub-spaces of a space with torsion, Proc. London Math. Soc. (2) 34 (1932), no. 1, 27-50. https://doi.org/10.1112/plms/s2-34.1.27
  15. D. H. Jin, Half lightlike submanifolds of a semi-Riemannian space form with a semisymmetric non-metric connection, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 1, 39-50.
  16. D. H. Jin and J. W. Lee, Einstein half lightlike submanifolds of a Lorentzian space form with a semi-symmetric metric connection, Quaest. Math. 37 (2014), no. 4, 485-505. https://doi.org/10.2989/16073606.2014.981686
  17. H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. of Math. (2) 27 (1925), no. 2, 91-98. https://doi.org/10.2307/1967964
  18. R. S. Mishra and S. N. Pandey, Semi-symmetric metric connections in an almost contact manifold, Indian J. Pure Appl. Math. 9 (1978), no. 6, 570-580.
  19. C. Murathan and C. Ozgur, Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions, Proc. Est. Acad. Sci. 57 (2008), no. 4, 210-216. https://doi.org/10.3176/proc.2008.4.02
  20. E. Pak, On the pseudo-Riemannian spaces, J. Korean Math. Soc. 6 (1969), 23-31.
  21. G. P. Pokhariyal and R. S. Mishra, Curvature tensors and their relativistic significance. II, Yokohama Math. J. 19 (1971), no. 2, 97-103.
  22. G. P. Pokhariyal, S. Yadav, and S. K. Chaubey, Ricci solitons on trans-Sasakian manifolds, Differ. Geom. Dyn. Syst. 20 (2018), 138-158.
  23. R. Sharma, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci. 12 (1989), no. 4, 787-790. https://doi.org/10.1155/S0161171289000967
  24. R. Sharma, Second order parallel tensors on contact manifolds, Algebras Groups Geom. 7 (1990), no. 2, 145-152.
  25. R. Sharma, Second order parallel tensors on contact manifolds. II, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), no. 6, 259-264.
  26. R. Sharma, On the curvature of contact metric manifolds, J. Geom. 53 (1995), no. 1-2, 179-190. https://doi.org/10.1007/BF01224050
  27. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y )${\cdot}$R = 0. I. The local version, J. Differential Geom. 17 (1982), no. 4, 531-582 (1983).
  28. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y )${\cdot}$R = 0. II. Global versions, Geom. Dedicata 19 (1985), no. 1, 65-108. https://doi.org/10.1007/BF00233102
  29. L. Tamassy and T. Q. Binh, On weak symmetries of Einstein and Sasakian manifolds, Tensor (N.S.) 53 (1993), Commemoration Volume I, 140-148.
  30. H. Weyl, Reine Infinitesimalgeometrie, Math. Z. 2 (1918), no. 3-4, 384-411. https://doi.org/10.1007/BF01199420
  31. K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15 (1970), 1579-1586.
  32. K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200. https://doi.org/10.3792/pia/1195579139
  33. K. Yano and S. Bochner, Curvature and Betti Numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, NJ, 1953.
  34. K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Differential Geometry 2 (1968), 161-184. https://doi.org/10.4310/jdg/1214428253
  35. F. Zengin, S. A. Demirbag, S. A. Uysal, and H. B. Yilmaz, Some vector fields on a Riemannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc. 38 (2012), no. 2, 479-490.