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(k,μ)-Paracontact Manifolds and Their Curvature Classification

Year 2022, Volume 43, Issue 3, 460 - 467, 30.09.2022
https://doi.org/10.17776/csj.1108962

Abstract

The aim of this paper is to study (k,μ)-Paracontact metric manifold. We introduce the curvature tensors of a (k,μ)-paracontact metric manifold satisfying the conditions R⋅P_*=0, R⋅L=0, R⋅W_1=0, R⋅W_0=0 and R⋅M=0. According to these cases, (k,μ)-paracontact manifolds have been characterized such as η-Einstein and Einstein. We get the necessary and sufficient conditions of a (k,μ)-paracontact metric manifold to be η-Einstein. Also, we consider new conclusions of a (k,μ)-paracontact metric manifold contribute to geometry. We think that some interesting results on a (k,μ)-paracontact metric manifold are obtained.

References

  • [1] Kaneyuki S., Williams F.L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985) 173-187.
  • [2] Zamkovoy S., Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36 (2009) 37-60.
  • [3] Cappelletti-Montano B., Küpeli Erken I., Murathan C., Nullity conditions in paracontact geometry, Differential Geom. Appl., 30 (2012) 665-693.
  • [4] Blair D.E., Koufogiorgos T., Papatoniou B.J., Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91 (1995) 189-214.
  • [5] Kowalezyk D., On some subclass of semi-symmetric manifolds, Soochow J. Math., 27 (2001) 445-461.
  • [6] Prasad B., A pseudo projective curvature tensor on a Riemannian manifold, Bull. Calcutta Math. Soc., 94 (3) (2002) 163-166.
  • [7] Ivanov S., Vassilev D., Zamkovoy S., Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata, 144 (2010) 79-100.
  • [8] Mert T., Characterization of some special curvature tensor on Almost C(a)-manifold, Asian Journal of Math. and Computer Research, 29 (1) (2022) 27-41.
  • [9] Mert T., Atçeken M., Almost C(a)-manifold on W_0^⋆-curvature tensor, Applied Mathematical Sciences, 15 (15) (2021) 693-703.
  • [10] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  • [11] [Boothby W.M., An Introduction to Differentiable Manifolds and Riemanniann Geometry, Academic Press, Inc. London, 1986.
  • [12] Zamkovoy S., Tzanov V., Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform., 100 (2011) 27-34.
  • [13] Ishii Y., On conharmonic transformations, Tensor N. S., 7 (1957) 73-80.
  • [14] Pokhariyal G.P., Mishra R.S., Curvature tensors and their relativistic significance II, Yokohama Math. J., 19 (2) (1971) 97-103.
  • [15] Atçeken M., Uygun P., Characterizations for totally geodesic submanifolds of (k,μ)-paracontact metric manifolds, Korean J. Math., 28 (2020) 555-571.
  • [16] Calvaruso G., Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55 (2011) 697-718.
  • [17] Szabo Z.I., Structure theorems on Riemannian spaces satisfying R(X,Y)⋅R=0, I: The local version, J. Differential Geom., 17 (4) (1982) 531-582.
  • [18] Tripathi M.M., Gupta P., T-curvature tensor on a semi-Riemannian manifold, J. Adv. Math. Studies., 4 (2011), No. 1, 117-129.
  • [19] Uygun P., Atçeken M., On (k,μ)-paracontact metric spaces satisfying some conditions on the W_0^⋆-curvature tensor, NTMSCI, 9(2) (2021) 26-37.

Year 2022, Volume 43, Issue 3, 460 - 467, 30.09.2022
https://doi.org/10.17776/csj.1108962

Abstract

References

  • [1] Kaneyuki S., Williams F.L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985) 173-187.
  • [2] Zamkovoy S., Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36 (2009) 37-60.
  • [3] Cappelletti-Montano B., Küpeli Erken I., Murathan C., Nullity conditions in paracontact geometry, Differential Geom. Appl., 30 (2012) 665-693.
  • [4] Blair D.E., Koufogiorgos T., Papatoniou B.J., Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91 (1995) 189-214.
  • [5] Kowalezyk D., On some subclass of semi-symmetric manifolds, Soochow J. Math., 27 (2001) 445-461.
  • [6] Prasad B., A pseudo projective curvature tensor on a Riemannian manifold, Bull. Calcutta Math. Soc., 94 (3) (2002) 163-166.
  • [7] Ivanov S., Vassilev D., Zamkovoy S., Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata, 144 (2010) 79-100.
  • [8] Mert T., Characterization of some special curvature tensor on Almost C(a)-manifold, Asian Journal of Math. and Computer Research, 29 (1) (2022) 27-41.
  • [9] Mert T., Atçeken M., Almost C(a)-manifold on W_0^⋆-curvature tensor, Applied Mathematical Sciences, 15 (15) (2021) 693-703.
  • [10] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  • [11] [Boothby W.M., An Introduction to Differentiable Manifolds and Riemanniann Geometry, Academic Press, Inc. London, 1986.
  • [12] Zamkovoy S., Tzanov V., Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform., 100 (2011) 27-34.
  • [13] Ishii Y., On conharmonic transformations, Tensor N. S., 7 (1957) 73-80.
  • [14] Pokhariyal G.P., Mishra R.S., Curvature tensors and their relativistic significance II, Yokohama Math. J., 19 (2) (1971) 97-103.
  • [15] Atçeken M., Uygun P., Characterizations for totally geodesic submanifolds of (k,μ)-paracontact metric manifolds, Korean J. Math., 28 (2020) 555-571.
  • [16] Calvaruso G., Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55 (2011) 697-718.
  • [17] Szabo Z.I., Structure theorems on Riemannian spaces satisfying R(X,Y)⋅R=0, I: The local version, J. Differential Geom., 17 (4) (1982) 531-582.
  • [18] Tripathi M.M., Gupta P., T-curvature tensor on a semi-Riemannian manifold, J. Adv. Math. Studies., 4 (2011), No. 1, 117-129.
  • [19] Uygun P., Atçeken M., On (k,μ)-paracontact metric spaces satisfying some conditions on the W_0^⋆-curvature tensor, NTMSCI, 9(2) (2021) 26-37.

Details

Primary Language English
Subjects Mathematics
Journal Section Natural Sciences
Authors

Pakize UYGUN> (Primary Author)
TOKAT GAZİOSMANPAŞA ÜNİVERSİTESİ
0000-0001-8226-4269
Türkiye

Publication Date September 30, 2022
Application Date April 25, 2022
Acceptance Date August 1, 2022
Published in Issue Year 2022, Volume 43, Issue 3

Cite

APA Uygun, P. (2022). (k,μ)-Paracontact Manifolds and Their Curvature Classification . Cumhuriyet Science Journal , 43 (3) , 460-467 . DOI: 10.17776/csj.1108962