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Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection

Year 2020, Volume: 3 Issue: 2, 78 - 85, 22.06.2020

S. K. Chaubey , Pankaj R. B. Kanaujia , S. K. Yadav

https://doi.org/10.32323/ujma.650209
https://izlik.org/JA43JJ38BY

Abstract

The aim of the present paper is to study the properties of Riemannian manifolds equipped with a projective semi-symmetric connection.

Keywords

Riemannian manifolds Projective semi-symmetric connection Different curvature tensors nullity distribution quasi Einstein manifolds

References

  • [1] A. A. Shaikh, Young Ho Kim and S. K. Hui, On Lorentzian quasi-Einstein manifolds, J. Korean Math. Soc., Vol. 48 (4), (2011), 669-689.
  • [2] A. Friedmann and J. A. Schouten, ¨U ber die geometrie der halbsymmetrischen ¨ ubertragung, Math. Zeitschr, 21, (1924), pp. 211-223.
  • [3] A. Kumar and S. K. Chaubey, A semi-symmetric non-metric connection in a generalized co-symplectic manifold, Int. Journal of Math. Analysis, 4(17), (2010), 809-817.
  • [4] A. K. Dubey, S. K. Chaubey and R. H. Ojha, On semi-symmetric non-metric connection, International Mathematical Forum, 5 (15), (2010), 731-737.
  • [5] A. Sharfuddin and S. I. Husain, Semi-symmetric metric connections in almost contact manifolds, Tensor N. S., 30 (1976), 133-139.
  • [6] B. O’ Neill, Semi-Riemannian Geometry with Applications to the Relativity, Academic Press, New York- London, 1983.
  • [7] D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
  • [8] ´E . Cartan, Sur une classe remarquable d’espaces de Riemannian, Bull. Soc. Math. France, 54 (1926), 214-264.
  • [9] ´E . Cartan, Le cons sur la geometrie des espaces de Riemann, 2nd ed., Paris, 1946.
  • [10] H. A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc., 34, (1932), pp. 27-50.
  • [11] H. Stephani, General relativity-An introduction to the theory of gravitational field, Cambridge Univ. Press, Cambridge, 1982.
  • [12] J. V. Narlikar, General relativity and gravitation, The Macmillan co. of India, 1978.
  • [13] K. Yano, On semi-symmetric metric connection, Rev. Roumaine de Math. Pures et Appl., 15 (1970), 1579-1586.
  • [14] M. C. Chaki and R. K. Maithy, On quasi Einstein manifolds, Publ. Math. Debrecen, 57, no. 3-4 (2000), 297-306.
  • [15] M. Okumura, Some remarks on space with a certain contact structure, Tohoku Math. J., 14 (1962), 135-145.
  • [16] M. C. Chaki, On generalized quasi Einstein manifolds, Publ. Math. Debrecen, 58, 4 (2001), 683-691.
  • [17] M. M. Tripathi and J. S. Kim, On N(k)-quasi Einstein manifolds, Commun. Korean Math. Soc., 22, no. 3 (2007), 411-417.
  • [18] N. S. Agashe and M. R. Chafle, A semi-symmetric non-metric connection in a Riemannian manifold, Indian J. Pure Appl. Math., 23 (1992), 399-409.
  • [19] O. Bahadır and S. K. Chaubey, Some notes on LP-Sasakian Manifolds with Generalized Symmetric Metric Connection, Honam Mathematical Journal (2020) (Accepted) arXiv:1805.00810v2 [math.DG] 17 Oct 2019.
  • [20] P. Alegre, David E. Blair and A. Carriazo, Generalized Sasakian Space forms, Israel J. of Math, 141 (2004), 157-183.
  • [21] P. Zhao and H. Song, An invariant of the projective semi-symmetric connection, Chinese Quarterly J. of Math., 16 (4), (2001), 48-52.
  • [22] P. Zhao, Some properties of projective semi-symmetric connection, Int. Math. Forum, 3(7), (2008), 341-347.
  • [23] P. Debnath and A. Konar, On quasi Einstein manifold and quasi Einstein spacetime, Differential Geometry-Dynamical Systems, 12 (2010), 73-82.
  • [24] Pankaj, S. K. Chaubey and R. Prasad, Trans-Sasakian Manifolds with respect to a non-symmetric non-metric connection, Global Journal of Advanced Research on Classical and Modern Geometries, Vol.7, (2018), Issue 1, 1-10.
  • [25] R. N. Singh, S. K. Pandey and Giteshwari Pandey, On semi-symmetric metric connection in an SP-Sasakian manifold, Proc. of the Nat. Academy of Sci., 83 (1) (2013), 39-47.
  • [26] R. Prasad and Pankaj, Some curvature tensors on a trans-Sasakian manifold with respect to semi symmetric non-metric connection, J. Nat. Acad. Math, Sp. Volume (2009), 55-64.
  • [27] R. S. Mishra and S. N. Pandey, Semi-symmetric metric connections in an almost contact manifold, Indian J. Pure Appl. Math., 9, No. 6 (1978), 570-580.
  • [28] S. Guha, On quasi Einstein and generalized quasi Einstein manifolds, Facta Univ. Ser. Mech. Automat. Control Robot, 3, 14 (2003), 821-842.
  • [29] S. K. Pal, M. K. Pandey and R. N. Singh, On a type of projective semi-symmetric connection, Int. J. of Anal. and Appl., 7(2) (2015), 153-161.
  • [30] S. K. Chaubey and A. Kumar, Semi-symmetric metric T-connection in an almost contact metric manifold, International Mathematical Forum, 5, (2010), no. 23, 1121 - 1129.
  • [31] S. K. Chaubey and R. H. Ojha, On a semi-symmetric non-metric and quarter-symmetric metric connections, Tensor N. S., 70 (2), (2008), 202-213.
  • [32] S. K. Chaubey and R. H. Ojha, On a semi-symmetric non-metric connection, Filomat, 26 (2), (2012), 269-275.
  • [33] S. K. Chaubey, Almost contact metric manifolds admitting semi-symmetric non-metric connection, Bulletin of Mathematical Analysis and Applications, 3(2), (2011), 252-260.
  • [34] S. K. Chaubey and A. C. Pandey, Some Properties of a Semi-symmetric Non-metric Connection on a Sasakian Manifold, Int. J. Contemp. Math. Sciences, 8(16) (2013), 789-799.
  • [35] S. K. Chaubey, Existence of N(k)-quasi Einstein manifolds, Facta Universitatis (NIS) Ser. Math. Inform., 32 (3), (2017), 369-385.
  • [36] S. K. Chaubey and S. K. Yadav, Study of Kenmotsu manifolds with semi-symmetric metric connection, Universal Journal of Mathematics and Applications, 1 (2) (2018), 89-97.
  • [37] S. K. Chaubey and U. C. De, Characterization of the Lorentzian para-Sasakian manifolds admitting a quarter-symmetric non-metric connection, SUT Journal of Mathematics, 55 (1), (2019), 53-67.
  • [38] S. K. Chaubey and A. Yildiz, Riemannian manifolds admitting a new type of semi symmetric nonmetric connection, Turk. J. Math., 43, (2019), 1887-1904.
  • [39] S. K. Chaubey and U. C. De, Lorentzian para-Sasakian manifolds admitting a new type of quarter symmetric non-metric x -connection, International Electronic Journal of Geometry, 12 (2), (2019), 266-275.
  • [40] S. K. Chaubey, J. W. Lee and S. Yadav, Riemannian manifolds with a semi-symmetric metric Pconnection J. Korean Math. Soc., 56 (4), (2019), 1113-1129.
  • [41] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441-417.
  • [42] S. Yadav, S. K. Chaubey and R. Prasad, On Kenmotsu manifolds with a semi-symmetric metric connection, Facta Universitatis (NIS) Ser. Math. Inform., Vol. 35, No.-1 (2020), 101-119.
  • [43] T. Imai, Notes on semi-symmetric metric connections, Tensor N. S., 24 (1972), 293-296.
  • [44] T. Takahashi, Sasakian f-symmetric spaces, Tohoku Math. J., 29 (1977), 91–113.
  • [45] U. C. De and B. K. De, On Quasi Einstein Manifolds, Commun. Korean Math. Soc. 23 (2008), No. 3, pp. 413–420.
  • [46] U. C. De and G. C. Ghosh, On quasi Einstein and special quasi Einstein manifolds, Proc. of the Int. conf. of Mathematics and its applications, Kuwait University, April 5-7, (2004), 178-191.
  • [47] U. C. De and G. C. Ghosh, On quasi Einstein manifolds, Period. Math. Hungar., 48, (1-2) (2004), 223-231.
  • [48] U. C. De and J. Sengupta, On a type of semi-symmetric metric connection on an almost contact metric manifold, Filomat, 14 (2000), 33-42.
  • [49] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X;Y)R = 0 I, The local version, J. Differ. Geometry 17 (1982), 531-582.
  • [50] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X;Y)R = 0 II, Global version, Geom. Dedicata 19 (1985), 65-108.

Year 2020, Volume: 3 Issue: 2, 78 - 85, 22.06.2020

S. K. Chaubey , Pankaj R. B. Kanaujia , S. K. Yadav

https://doi.org/10.32323/ujma.650209
https://izlik.org/JA43JJ38BY

Abstract

References

  • [1] A. A. Shaikh, Young Ho Kim and S. K. Hui, On Lorentzian quasi-Einstein manifolds, J. Korean Math. Soc., Vol. 48 (4), (2011), 669-689.
  • [2] A. Friedmann and J. A. Schouten, ¨U ber die geometrie der halbsymmetrischen ¨ ubertragung, Math. Zeitschr, 21, (1924), pp. 211-223.
  • [3] A. Kumar and S. K. Chaubey, A semi-symmetric non-metric connection in a generalized co-symplectic manifold, Int. Journal of Math. Analysis, 4(17), (2010), 809-817.
  • [4] A. K. Dubey, S. K. Chaubey and R. H. Ojha, On semi-symmetric non-metric connection, International Mathematical Forum, 5 (15), (2010), 731-737.
  • [5] A. Sharfuddin and S. I. Husain, Semi-symmetric metric connections in almost contact manifolds, Tensor N. S., 30 (1976), 133-139.
  • [6] B. O’ Neill, Semi-Riemannian Geometry with Applications to the Relativity, Academic Press, New York- London, 1983.
  • [7] D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
  • [8] ´E . Cartan, Sur une classe remarquable d’espaces de Riemannian, Bull. Soc. Math. France, 54 (1926), 214-264.
  • [9] ´E . Cartan, Le cons sur la geometrie des espaces de Riemann, 2nd ed., Paris, 1946.
  • [10] H. A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc., 34, (1932), pp. 27-50.
  • [11] H. Stephani, General relativity-An introduction to the theory of gravitational field, Cambridge Univ. Press, Cambridge, 1982.
  • [12] J. V. Narlikar, General relativity and gravitation, The Macmillan co. of India, 1978.
  • [13] K. Yano, On semi-symmetric metric connection, Rev. Roumaine de Math. Pures et Appl., 15 (1970), 1579-1586.
  • [14] M. C. Chaki and R. K. Maithy, On quasi Einstein manifolds, Publ. Math. Debrecen, 57, no. 3-4 (2000), 297-306.
  • [15] M. Okumura, Some remarks on space with a certain contact structure, Tohoku Math. J., 14 (1962), 135-145.
  • [16] M. C. Chaki, On generalized quasi Einstein manifolds, Publ. Math. Debrecen, 58, 4 (2001), 683-691.
  • [17] M. M. Tripathi and J. S. Kim, On N(k)-quasi Einstein manifolds, Commun. Korean Math. Soc., 22, no. 3 (2007), 411-417.
  • [18] N. S. Agashe and M. R. Chafle, A semi-symmetric non-metric connection in a Riemannian manifold, Indian J. Pure Appl. Math., 23 (1992), 399-409.
  • [19] O. Bahadır and S. K. Chaubey, Some notes on LP-Sasakian Manifolds with Generalized Symmetric Metric Connection, Honam Mathematical Journal (2020) (Accepted) arXiv:1805.00810v2 [math.DG] 17 Oct 2019.
  • [20] P. Alegre, David E. Blair and A. Carriazo, Generalized Sasakian Space forms, Israel J. of Math, 141 (2004), 157-183.
  • [21] P. Zhao and H. Song, An invariant of the projective semi-symmetric connection, Chinese Quarterly J. of Math., 16 (4), (2001), 48-52.
  • [22] P. Zhao, Some properties of projective semi-symmetric connection, Int. Math. Forum, 3(7), (2008), 341-347.
  • [23] P. Debnath and A. Konar, On quasi Einstein manifold and quasi Einstein spacetime, Differential Geometry-Dynamical Systems, 12 (2010), 73-82.
  • [24] Pankaj, S. K. Chaubey and R. Prasad, Trans-Sasakian Manifolds with respect to a non-symmetric non-metric connection, Global Journal of Advanced Research on Classical and Modern Geometries, Vol.7, (2018), Issue 1, 1-10.
  • [25] R. N. Singh, S. K. Pandey and Giteshwari Pandey, On semi-symmetric metric connection in an SP-Sasakian manifold, Proc. of the Nat. Academy of Sci., 83 (1) (2013), 39-47.
  • [26] R. Prasad and Pankaj, Some curvature tensors on a trans-Sasakian manifold with respect to semi symmetric non-metric connection, J. Nat. Acad. Math, Sp. Volume (2009), 55-64.
  • [27] R. S. Mishra and S. N. Pandey, Semi-symmetric metric connections in an almost contact manifold, Indian J. Pure Appl. Math., 9, No. 6 (1978), 570-580.
  • [28] S. Guha, On quasi Einstein and generalized quasi Einstein manifolds, Facta Univ. Ser. Mech. Automat. Control Robot, 3, 14 (2003), 821-842.
  • [29] S. K. Pal, M. K. Pandey and R. N. Singh, On a type of projective semi-symmetric connection, Int. J. of Anal. and Appl., 7(2) (2015), 153-161.
  • [30] S. K. Chaubey and A. Kumar, Semi-symmetric metric T-connection in an almost contact metric manifold, International Mathematical Forum, 5, (2010), no. 23, 1121 - 1129.
  • [31] S. K. Chaubey and R. H. Ojha, On a semi-symmetric non-metric and quarter-symmetric metric connections, Tensor N. S., 70 (2), (2008), 202-213.
  • [32] S. K. Chaubey and R. H. Ojha, On a semi-symmetric non-metric connection, Filomat, 26 (2), (2012), 269-275.
  • [33] S. K. Chaubey, Almost contact metric manifolds admitting semi-symmetric non-metric connection, Bulletin of Mathematical Analysis and Applications, 3(2), (2011), 252-260.
  • [34] S. K. Chaubey and A. C. Pandey, Some Properties of a Semi-symmetric Non-metric Connection on a Sasakian Manifold, Int. J. Contemp. Math. Sciences, 8(16) (2013), 789-799.
  • [35] S. K. Chaubey, Existence of N(k)-quasi Einstein manifolds, Facta Universitatis (NIS) Ser. Math. Inform., 32 (3), (2017), 369-385.
  • [36] S. K. Chaubey and S. K. Yadav, Study of Kenmotsu manifolds with semi-symmetric metric connection, Universal Journal of Mathematics and Applications, 1 (2) (2018), 89-97.
  • [37] S. K. Chaubey and U. C. De, Characterization of the Lorentzian para-Sasakian manifolds admitting a quarter-symmetric non-metric connection, SUT Journal of Mathematics, 55 (1), (2019), 53-67.
  • [38] S. K. Chaubey and A. Yildiz, Riemannian manifolds admitting a new type of semi symmetric nonmetric connection, Turk. J. Math., 43, (2019), 1887-1904.
  • [39] S. K. Chaubey and U. C. De, Lorentzian para-Sasakian manifolds admitting a new type of quarter symmetric non-metric x -connection, International Electronic Journal of Geometry, 12 (2), (2019), 266-275.
  • [40] S. K. Chaubey, J. W. Lee and S. Yadav, Riemannian manifolds with a semi-symmetric metric Pconnection J. Korean Math. Soc., 56 (4), (2019), 1113-1129.
  • [41] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441-417.
  • [42] S. Yadav, S. K. Chaubey and R. Prasad, On Kenmotsu manifolds with a semi-symmetric metric connection, Facta Universitatis (NIS) Ser. Math. Inform., Vol. 35, No.-1 (2020), 101-119.
  • [43] T. Imai, Notes on semi-symmetric metric connections, Tensor N. S., 24 (1972), 293-296.
  • [44] T. Takahashi, Sasakian f-symmetric spaces, Tohoku Math. J., 29 (1977), 91–113.
  • [45] U. C. De and B. K. De, On Quasi Einstein Manifolds, Commun. Korean Math. Soc. 23 (2008), No. 3, pp. 413–420.
  • [46] U. C. De and G. C. Ghosh, On quasi Einstein and special quasi Einstein manifolds, Proc. of the Int. conf. of Mathematics and its applications, Kuwait University, April 5-7, (2004), 178-191.
  • [47] U. C. De and G. C. Ghosh, On quasi Einstein manifolds, Period. Math. Hungar., 48, (1-2) (2004), 223-231.
  • [48] U. C. De and J. Sengupta, On a type of semi-symmetric metric connection on an almost contact metric manifold, Filomat, 14 (2000), 33-42.
  • [49] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X;Y)R = 0 I, The local version, J. Differ. Geometry 17 (1982), 531-582.
  • [50] Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X;Y)R = 0 II, Global version, Geom. Dedicata 19 (1985), 65-108.
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

S. K. Chaubey 0000-0002-3882-4596

Pankaj R. B. Kanaujia 0000-0002-3313-6006

S. K. Yadav

Submission Date November 23, 2019
Acceptance Date April 14, 2020
Publication Date June 22, 2020
DOI https://doi.org/10.32323/ujma.650209
IZ https://izlik.org/JA43JJ38BY
Published in Issue Year 2020 Volume: 3 Issue: 2

Cite

APA Chaubey, S. K., R. B. Kanaujia, P., & Yadav, S. K. (2020). Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection. Universal Journal of Mathematics and Applications, 3(2), 78-85. https://doi.org/10.32323/ujma.650209
AMA 1.Chaubey SK, R. B. Kanaujia P, Yadav SK. Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection. Univ. J. Math. Appl. 2020;3(2):78-85. doi:10.32323/ujma.650209
Chicago Chaubey, S. K., Pankaj R. B. Kanaujia, and S. K. Yadav. 2020. “Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection”. Universal Journal of Mathematics and Applications 3 (2): 78-85. https://doi.org/10.32323/ujma.650209.
EndNote Chaubey SK, R. B. Kanaujia P, Yadav SK (June 1, 2020) Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection. Universal Journal of Mathematics and Applications 3 2 78–85.
IEEE [1]S. K. Chaubey, P. R. B. Kanaujia, and S. K. Yadav, “Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection”, Univ. J. Math. Appl., vol. 3, no. 2, pp. 78–85, June 2020, doi: 10.32323/ujma.650209.
ISNAD Chaubey, S. K. - R. B. Kanaujia, Pankaj - Yadav, S. K. “Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection”. Universal Journal of Mathematics and Applications 3/2 (June 1, 2020): 78-85. https://doi.org/10.32323/ujma.650209.
JAMA 1.Chaubey SK, R. B. Kanaujia P, Yadav SK. Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection. Univ. J. Math. Appl. 2020;3:78–85.
MLA Chaubey, S. K., et al. “Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection”. Universal Journal of Mathematics and Applications, vol. 3, no. 2, June 2020, pp. 78-85, doi:10.32323/ujma.650209.
Vancouver 1.S. K. Chaubey, Pankaj R. B. Kanaujia, S. K. Yadav. Projective Curvature Tensor of Riemannian Manifolds Admitting A Projective Semi-Symmetric Connection. Univ. J. Math. Appl. 2020 Jun. 1;3(2):78-85. doi:10.32323/ujma.650209

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