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ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES

Year 2012, Volume: 5 Issue: 1, 42 - 77, 30.04.2012

Mukut Mani Trıpathı , Punam Gupta

Abstract




Keywords

(N (k) ξ)-semi-Riemannian manifold T -curvature tensor quasi- conformal curvature tensor conformal curvature tensor conharmonic curvature tensor con- circular curvature tensor pseudo-projective curvature tensor projective curvature tensor M- projective curvature tensor

References

  • [1] Adati, T. and Matsumoto, K., On conformally recurrent and conformally symmetric P - Sasakian manifolds, TRU Math., 13 (1977), 25–32.
  • [2] Adati, T. and Miyazawa, T., On P -Sasakian manifolds admitting some parallel and recurrent tensors, Tensor (N.S.), 33 (1979), no. 3, 287–292.
  • [3] Beem, J.K. and Ehrlich, P.E., Global Lorentzian geometry, Marcel Dekker, New York, 1981.
  • [4] Blair, D.E., Contact manifolds in Riemannian geometry, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 509 (1976), 146.
  • [5] Blair, D.E., Kim, J.-S. and Tripathi, M.M., On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42 (2005), no. 5, 883–892.
  • [6] Chaki, M.C. and Tarafdar, M., On a type of Sasakian manifold, Soochow J. Math., 16 (1990), no. 1, 23–28.
  • [7] De, U.C. and Biswas, S., A note on ξ -conformally flat contact manifolds, Bull. Malays. Math. Sci. Soc. (2), 29 (2006), no. 1, 51–57.
  • [8] De, U.C. and Ghosh, J.C., On a type of contact manifold, Note Mat., 14 (1994), no. 2, 155–160 (1997).
  • [9] De, U.C., Jun, J.B. and Gazi, A.K., Sasakian manifolds with quasi-conformal curvature tensor, Bull. Korean Math. Soc., 45 (2008), no. 2, 313–319.
  • [10] Deszcz, R., On Ricci-pseudosymmetric warped products, Demonstratio Math., 22 (1989), 1053–1065.
  • [11] Duggal, K.L., Space time manifolds and contact structures, Internat. J. Math. Math. Sci., 13 (1990), 545–554.
  • [12] Dwivedi, M.K. and Kim, J.-S., On conharmonic curvature tensor in K-contact and Sasakian manifolds, Bull. Malays. Math. Sci. Soc., 34 (2011), no. 1, 171–180.
  • [13] Eisenhart, L.P., Riemannian Geometry, Princeton University Press, 1949. [14] Ishii, Y., On conharmonic transformations, Tensor (N.S.), 7 (1957), 73–80.
  • [15] Ghosh, A. and Sharma, R., Some results on contact metric manifolds, Ann. Global Anal. Geom., 15 (1997), no. 6, 497–507.
  • [16] Hong, S., Özgür, C. and Tripathi, M.M., On some classes of Kenmotsu manifold, Kuwait J. Sci. Engrg., 33 (2006), no. 2, 19–32.
  • [17] Jun, J.-B., De, U.C. and Pathak, G., On Kenmotsu manifolds, J. Korean Math. Soc., 42 (2005), no. 3, 435–445.
  • [18] Kenmotsu, K., A class of almost contact Riemannian manifold, Tˆohoku Math. J. (2), 24 (1972), 93–103.
  • [19] Koufogiorgos, T., Contact metric manifolds, Ann. Global Anal. Geom., 11 (1993), no. 1, 25–34.
  • [20] Koufogiorgos, T., Contact Riemannian manifolds with constant φ-sectional curvature, Tokyo J. Math., 20 (1997), no. 1, 13–22.
  • [21] Maralabhavi, Y.B., On W -symmetric and W -recurrent Sasakian manifolds, J. Nat. Acad. Math. India, 4 (1986), no. 1-2, 63–72.
  • [22] Matsumoto, K., On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Sci., 12 (1989), no. 2, 151–156.
  • [23] Miyazawa, T. and Yamaguchi, S., Some theorems on K-contact metric manifolds and Sasakian manifolds, TRU Math., 2 (1966), 46–52.
  • [24] Mishra, R.S., On Sasakian manifolds, Indian J. Pure Appl. Math., 1 (1970), no. 1, 98–105.
  • [25] Mishra, I.K. and Ojha, R.H., On para Sasakian manifolds, Indian J. Pure Appl. Math., 32 (2001), no. 9, 1309–1316.
  • [26] Miyazawa, T. and Yamaguchi, S., Some theorems on K-contact metric manifolds and Sasakian manifolds, TRU Math., 2 (1966), 46–52.
  • [27] Nomizu, K. and Ozeki, H., A theorem on tensor fields, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 206–207.
  • [28] Okumura, M., Some remarks on space with a certain contact structure, Tˆohoku Math. J. (2), 14 (1962), 135–145.
  • [29] Ojha, R.H., M -projectively flat Sasakian manifolds, Indian J. Pure Appl. Math., 17 (1986), no. 4, 481–484.
  • [30] O’Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, London, 1983.
  • [31] Özgür, C., On a class of para-Sasakian manifold, Turk. J. Math., 29 (2005), 249–257.
  • [32] Özgür, C. and De, U.C., On the quasi-conformal curvature tensor of a Kenmotsu manifold, Math. Pannon., 17 (2006), no. 2, 221–228.
  • [33] Özgür, C. and Tripathi, M.M., On P -Sasakian manifolds satisfying certain conditions on the concircular curvature tensor, Turkish J. Math., 31 (2007), no. 2, 171–179.
  • [34] Pandey, S.N. and Verma, S., On para-Sasakian manifold, Indian J. Pure Appl. Math., 30 (1999), no. 1, 15–22.
  • [35] Perrone, D., Contact Riemannian manifolds satisfying R(X, ξ) · R = 0, Yokohama Math. J., 39 (1992), no. 2, 141–149.
  • [36] Pokhariyal, G.P. and Mishra, R.S., Curvature tensors and their relativistic significance, Yoko- hama Math. J., 18 (1970), 105–108.
  • [37] Pokhariyal, G.P. and Mishra, R.S., Curvature tensors and their relativistic significance II, Yokohama Math. J., 19 (1971), no. 2, 97–103.
  • [38] Pokhariyal, G.P., Relativistic significance of curvature tensors, Internat. J. Math. Math. Sci., 5 (1982), no. 1, 133–139.
  • [39] Prasad, B., A pseudo projective curvature tensor on a Riemannian manifold, Bull. Calcutta Math. Soc., 94 (2002), no. 3, 163–166.
  • [40] Rahman, M.S., A study of para-Sasakian manifolds, International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization, International Centre for Theoretical Physics, Internal Report, 1995.
  • [41] Sasaki, S., On differentiate manifolds with certain structures which are closely related to almost contact structure I, Tˆohoku Math. J., 12 (1960), 459–476.
  • [42] Satö, I., On a structure similar to the almost contact structure, Tensor (N.S.), 30 (1976), no.3, 219–224.
  • [43] Satö, I. and Matsumoto, K., On P -Sasakian manifold satisfying certain conditions, Tensor (N.S.), 33 (1979), 173–178.
  • [44] Sharfuddin, A., Deshmukh, S. and Husain, S.I., On para-Sasakian manifolds, Indian J. Pure Appl. Math., 11 (1980), no. 7, 845–853.
  • [45] Szabó, Z.I., Structure theorems on Riemannian manifolds satisfying R(X, Y ) · R = 0, I, Local version, J. Diff. Geom., 17 (1982), 531–582.
  • [46] Takagi, H., An example of Riemannian manifold satisfying R(X, Y ) · R = 0 but not ∇R = 0, Tˆohoku Math. J., 24 (1972), 105–108.
  • [47] Takahashi, T., Sasakian manifold with pseudo-Riemannian metric, Tˆohoku Math J., 21 (1969), 271–290.
  • [48] Tanno, S., Ricci curvatures of contact Riemannian manifolds, Tˆohoku Math J., 40 (1988), 441–448.
  • [49] Tarafdar, M. and Mayra, A., On P -Sasakian manifold, I˙stanb. U¨ niv. Fen Fak. Mat. Fiz. Astron. Derg., 53 (1994), 73–76 (1996).
  • [50] Tarafdar, M. and Sengupta, A.K., On a type of contact manifolds, Anal. Stint Ale Univ. “ Al. I. Cuza” Iasi, 47 (2001), no.1, 73–78 (2002).
  • [51] Tripathi, M.M., Kılı¸c, E., Yüksel Perkta¸s, S. and Kele¸s, S., Indefinite almost paracontact metric manifolds, Internat. J. Math. Math. Sci., 2010 (2010), 19 pp. Art. ID 846195.
  • [52] Tripathi, M.M. and Gupta, P., T -curvature tensor on a semi-Riemannian manifolds, Jour. Adv. Math. Stud., 4 (2011), no. 1, 117–129.
  • [53] Walker, A.G., On Ruse’s spaces of recurrent curvature, Proc. London Math. Soc., 50 (1950), 36–64.
  • [54] Yano, K., Concircular Geometry I. Concircular transformations, Math. Institute, Tokyo Im- perial Univ. Proc., 16 (1940), 195–200.
  • [55] Yano, K. and Bochner, S., Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, 1953.
  • [56] Yano, K. and Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom., 2 (1968), 161–184.

Year 2012, Volume: 5 Issue: 1, 42 - 77, 30.04.2012

Mukut Mani Trıpathı , Punam Gupta

Abstract

References

  • [1] Adati, T. and Matsumoto, K., On conformally recurrent and conformally symmetric P - Sasakian manifolds, TRU Math., 13 (1977), 25–32.
  • [2] Adati, T. and Miyazawa, T., On P -Sasakian manifolds admitting some parallel and recurrent tensors, Tensor (N.S.), 33 (1979), no. 3, 287–292.
  • [3] Beem, J.K. and Ehrlich, P.E., Global Lorentzian geometry, Marcel Dekker, New York, 1981.
  • [4] Blair, D.E., Contact manifolds in Riemannian geometry, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 509 (1976), 146.
  • [5] Blair, D.E., Kim, J.-S. and Tripathi, M.M., On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42 (2005), no. 5, 883–892.
  • [6] Chaki, M.C. and Tarafdar, M., On a type of Sasakian manifold, Soochow J. Math., 16 (1990), no. 1, 23–28.
  • [7] De, U.C. and Biswas, S., A note on ξ -conformally flat contact manifolds, Bull. Malays. Math. Sci. Soc. (2), 29 (2006), no. 1, 51–57.
  • [8] De, U.C. and Ghosh, J.C., On a type of contact manifold, Note Mat., 14 (1994), no. 2, 155–160 (1997).
  • [9] De, U.C., Jun, J.B. and Gazi, A.K., Sasakian manifolds with quasi-conformal curvature tensor, Bull. Korean Math. Soc., 45 (2008), no. 2, 313–319.
  • [10] Deszcz, R., On Ricci-pseudosymmetric warped products, Demonstratio Math., 22 (1989), 1053–1065.
  • [11] Duggal, K.L., Space time manifolds and contact structures, Internat. J. Math. Math. Sci., 13 (1990), 545–554.
  • [12] Dwivedi, M.K. and Kim, J.-S., On conharmonic curvature tensor in K-contact and Sasakian manifolds, Bull. Malays. Math. Sci. Soc., 34 (2011), no. 1, 171–180.
  • [13] Eisenhart, L.P., Riemannian Geometry, Princeton University Press, 1949. [14] Ishii, Y., On conharmonic transformations, Tensor (N.S.), 7 (1957), 73–80.
  • [15] Ghosh, A. and Sharma, R., Some results on contact metric manifolds, Ann. Global Anal. Geom., 15 (1997), no. 6, 497–507.
  • [16] Hong, S., Özgür, C. and Tripathi, M.M., On some classes of Kenmotsu manifold, Kuwait J. Sci. Engrg., 33 (2006), no. 2, 19–32.
  • [17] Jun, J.-B., De, U.C. and Pathak, G., On Kenmotsu manifolds, J. Korean Math. Soc., 42 (2005), no. 3, 435–445.
  • [18] Kenmotsu, K., A class of almost contact Riemannian manifold, Tˆohoku Math. J. (2), 24 (1972), 93–103.
  • [19] Koufogiorgos, T., Contact metric manifolds, Ann. Global Anal. Geom., 11 (1993), no. 1, 25–34.
  • [20] Koufogiorgos, T., Contact Riemannian manifolds with constant φ-sectional curvature, Tokyo J. Math., 20 (1997), no. 1, 13–22.
  • [21] Maralabhavi, Y.B., On W -symmetric and W -recurrent Sasakian manifolds, J. Nat. Acad. Math. India, 4 (1986), no. 1-2, 63–72.
  • [22] Matsumoto, K., On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Sci., 12 (1989), no. 2, 151–156.
  • [23] Miyazawa, T. and Yamaguchi, S., Some theorems on K-contact metric manifolds and Sasakian manifolds, TRU Math., 2 (1966), 46–52.
  • [24] Mishra, R.S., On Sasakian manifolds, Indian J. Pure Appl. Math., 1 (1970), no. 1, 98–105.
  • [25] Mishra, I.K. and Ojha, R.H., On para Sasakian manifolds, Indian J. Pure Appl. Math., 32 (2001), no. 9, 1309–1316.
  • [26] Miyazawa, T. and Yamaguchi, S., Some theorems on K-contact metric manifolds and Sasakian manifolds, TRU Math., 2 (1966), 46–52.
  • [27] Nomizu, K. and Ozeki, H., A theorem on tensor fields, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 206–207.
  • [28] Okumura, M., Some remarks on space with a certain contact structure, Tˆohoku Math. J. (2), 14 (1962), 135–145.
  • [29] Ojha, R.H., M -projectively flat Sasakian manifolds, Indian J. Pure Appl. Math., 17 (1986), no. 4, 481–484.
  • [30] O’Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, London, 1983.
  • [31] Özgür, C., On a class of para-Sasakian manifold, Turk. J. Math., 29 (2005), 249–257.
  • [32] Özgür, C. and De, U.C., On the quasi-conformal curvature tensor of a Kenmotsu manifold, Math. Pannon., 17 (2006), no. 2, 221–228.
  • [33] Özgür, C. and Tripathi, M.M., On P -Sasakian manifolds satisfying certain conditions on the concircular curvature tensor, Turkish J. Math., 31 (2007), no. 2, 171–179.
  • [34] Pandey, S.N. and Verma, S., On para-Sasakian manifold, Indian J. Pure Appl. Math., 30 (1999), no. 1, 15–22.
  • [35] Perrone, D., Contact Riemannian manifolds satisfying R(X, ξ) · R = 0, Yokohama Math. J., 39 (1992), no. 2, 141–149.
  • [36] Pokhariyal, G.P. and Mishra, R.S., Curvature tensors and their relativistic significance, Yoko- hama Math. J., 18 (1970), 105–108.
  • [37] Pokhariyal, G.P. and Mishra, R.S., Curvature tensors and their relativistic significance II, Yokohama Math. J., 19 (1971), no. 2, 97–103.
  • [38] Pokhariyal, G.P., Relativistic significance of curvature tensors, Internat. J. Math. Math. Sci., 5 (1982), no. 1, 133–139.
  • [39] Prasad, B., A pseudo projective curvature tensor on a Riemannian manifold, Bull. Calcutta Math. Soc., 94 (2002), no. 3, 163–166.
  • [40] Rahman, M.S., A study of para-Sasakian manifolds, International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization, International Centre for Theoretical Physics, Internal Report, 1995.
  • [41] Sasaki, S., On differentiate manifolds with certain structures which are closely related to almost contact structure I, Tˆohoku Math. J., 12 (1960), 459–476.
  • [42] Satö, I., On a structure similar to the almost contact structure, Tensor (N.S.), 30 (1976), no.3, 219–224.
  • [43] Satö, I. and Matsumoto, K., On P -Sasakian manifold satisfying certain conditions, Tensor (N.S.), 33 (1979), 173–178.
  • [44] Sharfuddin, A., Deshmukh, S. and Husain, S.I., On para-Sasakian manifolds, Indian J. Pure Appl. Math., 11 (1980), no. 7, 845–853.
  • [45] Szabó, Z.I., Structure theorems on Riemannian manifolds satisfying R(X, Y ) · R = 0, I, Local version, J. Diff. Geom., 17 (1982), 531–582.
  • [46] Takagi, H., An example of Riemannian manifold satisfying R(X, Y ) · R = 0 but not ∇R = 0, Tˆohoku Math. J., 24 (1972), 105–108.
  • [47] Takahashi, T., Sasakian manifold with pseudo-Riemannian metric, Tˆohoku Math J., 21 (1969), 271–290.
  • [48] Tanno, S., Ricci curvatures of contact Riemannian manifolds, Tˆohoku Math J., 40 (1988), 441–448.
  • [49] Tarafdar, M. and Mayra, A., On P -Sasakian manifold, I˙stanb. U¨ niv. Fen Fak. Mat. Fiz. Astron. Derg., 53 (1994), 73–76 (1996).
  • [50] Tarafdar, M. and Sengupta, A.K., On a type of contact manifolds, Anal. Stint Ale Univ. “ Al. I. Cuza” Iasi, 47 (2001), no.1, 73–78 (2002).
  • [51] Tripathi, M.M., Kılı¸c, E., Yüksel Perkta¸s, S. and Kele¸s, S., Indefinite almost paracontact metric manifolds, Internat. J. Math. Math. Sci., 2010 (2010), 19 pp. Art. ID 846195.
  • [52] Tripathi, M.M. and Gupta, P., T -curvature tensor on a semi-Riemannian manifolds, Jour. Adv. Math. Stud., 4 (2011), no. 1, 117–129.
  • [53] Walker, A.G., On Ruse’s spaces of recurrent curvature, Proc. London Math. Soc., 50 (1950), 36–64.
  • [54] Yano, K., Concircular Geometry I. Concircular transformations, Math. Institute, Tokyo Im- perial Univ. Proc., 16 (1940), 195–200.
  • [55] Yano, K. and Bochner, S., Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, 1953.
  • [56] Yano, K. and Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom., 2 (1968), 161–184.
There are 55 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Mukut Mani Trıpathı

Punam Gupta This is me

Publication Date April 30, 2012
Published in Issue Year 2012 Volume: 5 Issue: 1

Cite

APA Trıpathı, M. M., & Gupta, P. (2012). ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES. International Electronic Journal of Geometry, 5(1), 42-77.
AMA Trıpathı MM, Gupta P. ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES. Int. Electron. J. Geom. April 2012;5(1):42-77.
Chicago Trıpathı, Mukut Mani, and Punam Gupta. “ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES”. International Electronic Journal of Geometry 5, no. 1 (April 2012): 42-77.
EndNote Trıpathı MM, Gupta P (April 1, 2012) ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES. International Electronic Journal of Geometry 5 1 42–77.
IEEE M. M. Trıpathı and P. Gupta, “ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES”, Int. Electron. J. Geom., vol. 5, no. 1, pp. 42–77, 2012.
ISNAD Trıpathı, Mukut Mani - Gupta, Punam. “ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES”. International Electronic Journal of Geometry 5/1 (April 2012), 42-77.
JAMA Trıpathı MM, Gupta P. ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES. Int. Electron. J. Geom. 2012;5:42–77.
MLA Trıpathı, Mukut Mani and Punam Gupta. “ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES”. International Electronic Journal of Geometry, vol. 5, no. 1, 2012, pp. 42-77.
Vancouver Trıpathı MM, Gupta P. ON (N(k);ξ)-SEMI-RIEMANNIAN MANIFOLDS: SEMISYMMETRIES. Int. Electron. J. Geom. 2012;5(1):42-77.