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

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

doi:10.32323/ujma.650209

Research Article

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

Year 2020, , 78 - 85, 22.06.2020

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

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

Cited By: 2

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, , 78 - 85, 22.06.2020

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

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

Cited By: 2

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	Articles
Authors	S. K. Chaubey 0000-0002-3882-4596 Pankaj R. B. Kanaujia 0000-0002-3313-6006 S. K. Yadav
Publication Date	June 22, 2020
Submission Date	November 23, 2019
Acceptance Date	April 14, 2020
Published in Issue	Year 2020

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