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Year 2020, Volume: 13 Issue: 1, 94 - 106, 30.01.2020

Lovejoy Das , Haşim Çayır

https://doi.org/10.36890/iejg.559746

Abstract

References

  • [1] Andreou, F. G.: On integrability conditions of a structure $f$ satisfying $f^{5}+f=0$. Tensor N.S. 40, 27–31 (1983).
  • [2] Çayır, H.: Some Notes on Lifts of Almost Paracontact Structures. American Review of Mathematics and Statistics. 3(1), 52–60 (2015).
  • [3] Çayır, H.: Lie derivatives of almost contact structure and almost paracontact structure with respect to $X^{V}$ and $X^{H}$ on tangent bundle $T(M)$. Proceedings of the Institute of Mathematics and Mechanics. 42(1), 38–49 (2016).
  • [4] Çayır, H.: Tachibana and Vishnevskii Operators Applied to $X^{V}$\ and $X^{H}$ in Almost Paracontact Structure on Tangent Bundle $T(M)$. New Trends in Mathematical Sciences. 4(3), 105–115 (2016).
  • [5] Çayır, H., Köseoğlu, G.: Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure With Respect to $X^{C}$ and $X^{V}$. New Trends in Mathematical Sciences. 4(1), 153–159 (2016).
  • [6] Das, Lovejoy S.: On CR-structure and an $f(2K+4;2)-$ structure satisfying $f^{2K+4}+f^{2}=0$.Tensor. 73(3), 222–227 (2011).
  • [7] Das, Lovejoy S.: On lifts of structure satisfying $F^{K+1}-a^{2}F^{K-1}=0$. Kyungpook Mathematical Journal. 40(2), 391–398 (2000).
  • [8] Das, Lovejoy S.: Some problems on horizantal and complete lifts of $F((K+1)(K-1))-$structure ($K$, odd and $\geqslant 3$). Mathematica Balkanika. 7, 57–62 (1978).
  • [9] Das, Lovejoy S., Nivas, R., Pathak, V. N.: On horizontal and complete lifts from a manifold with $f\lambda (7,1)-$structure to its cotangent bundle. International Journal of Mathematics and Mathematical Sciences. 8, 1291–1297 (2005).
  • [10] Gupta, V.C.: Integrability Conditions of a Structure $F$ Satisfying $F^{K}+F=0$. The Nepali Math. Sc. Report. 14(2), 55-62 (1998).
  • [11] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry-Volume I. John Wiley & Sons Inc, New York (1963).
  • [12] Leon, Manuel de.: Existence and Integrability conditions of $\ \phi (k+1,k-1)$ structure on $(K+1)n-$dimensional manifolds. Rev. Roumaine Math. Pures Appl. 29, 479–489 (1984).
  • [13] Nivas, R., Prasad, C. S.: On a structure defined by a tensor field $f(\neq 0)$ of type $(1,1)$ satisfying $f^{5}-a^{2}f=0$. Nep. Math. Sc. Rep. 10(1), 25–30 (1985).
  • [14] Salimov, A. A.: Tensor Operators and Their applications. Nova Science Publ., New York (2013).
  • [15] Salimov, A. A., Çayır, H.: Some Notes On Almost Paracontact Structures. Comptes Rendus de l’Acedemie Bulgare Des Sciences. 66(3), 331-338 (2013).
  • [16] Singh, A.: On $CR-$structures $F-$structures satisfying $ F^{2K+P}+F^{P}=0$. Int. J. Contemp. Math. Sciences. 4, 1029–1035 (2009).
  • [17] Singh, A., Pandey, R. K., Khare, S.: {On horizontal and complete lifts of $(1,1)$ tensor fields $F$ satisfying the structure equation $F(2K+S,S)=0$. International Journal of Mathematics and Soft Computing. 6(1), 143–152 (2016).
  • [18] Yano, K., Patterson, E. M.: Horizontal lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan. 19, 185–198 (1967).
  • [19] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker Inc., New York (1973).
  • [20] Yano, K., Ishihara, S.: On integrabilitiy of a structure f satisfying $f^{3}+f=0$. Quart, J. Math. 25, 217–222 (1964).

On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle

Year 2020, Volume: 13 Issue: 1, 94 - 106, 30.01.2020

Lovejoy Das , Haşim Çayır

https://doi.org/10.36890/iejg.559746

Abstract

This paper consists of two main sections. In the first part, we find the integrability conditions of the horizontal lifts of $F((K+1),(K-1))-$ structure satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$. Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of $F((K+1),(K-1))-$structure in cotangent bundle $T^{\ast }(M^{n})$. Finally, we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of the structure. In the second part, all results obtained in the first section were obtained according to the complete and horizontal lifts of the structure in tangent bundle $T(M^{n})$.

Keywords

Integrability conditions Tachibana operators $CR-$Submanifolds $CR-$Stucture tangent bundle cotangent bundle

References

  • [1] Andreou, F. G.: On integrability conditions of a structure $f$ satisfying $f^{5}+f=0$. Tensor N.S. 40, 27–31 (1983).
  • [2] Çayır, H.: Some Notes on Lifts of Almost Paracontact Structures. American Review of Mathematics and Statistics. 3(1), 52–60 (2015).
  • [3] Çayır, H.: Lie derivatives of almost contact structure and almost paracontact structure with respect to $X^{V}$ and $X^{H}$ on tangent bundle $T(M)$. Proceedings of the Institute of Mathematics and Mechanics. 42(1), 38–49 (2016).
  • [4] Çayır, H.: Tachibana and Vishnevskii Operators Applied to $X^{V}$\ and $X^{H}$ in Almost Paracontact Structure on Tangent Bundle $T(M)$. New Trends in Mathematical Sciences. 4(3), 105–115 (2016).
  • [5] Çayır, H., Köseoğlu, G.: Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure With Respect to $X^{C}$ and $X^{V}$. New Trends in Mathematical Sciences. 4(1), 153–159 (2016).
  • [6] Das, Lovejoy S.: On CR-structure and an $f(2K+4;2)-$ structure satisfying $f^{2K+4}+f^{2}=0$.Tensor. 73(3), 222–227 (2011).
  • [7] Das, Lovejoy S.: On lifts of structure satisfying $F^{K+1}-a^{2}F^{K-1}=0$. Kyungpook Mathematical Journal. 40(2), 391–398 (2000).
  • [8] Das, Lovejoy S.: Some problems on horizantal and complete lifts of $F((K+1)(K-1))-$structure ($K$, odd and $\geqslant 3$). Mathematica Balkanika. 7, 57–62 (1978).
  • [9] Das, Lovejoy S., Nivas, R., Pathak, V. N.: On horizontal and complete lifts from a manifold with $f\lambda (7,1)-$structure to its cotangent bundle. International Journal of Mathematics and Mathematical Sciences. 8, 1291–1297 (2005).
  • [10] Gupta, V.C.: Integrability Conditions of a Structure $F$ Satisfying $F^{K}+F=0$. The Nepali Math. Sc. Report. 14(2), 55-62 (1998).
  • [11] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry-Volume I. John Wiley & Sons Inc, New York (1963).
  • [12] Leon, Manuel de.: Existence and Integrability conditions of $\ \phi (k+1,k-1)$ structure on $(K+1)n-$dimensional manifolds. Rev. Roumaine Math. Pures Appl. 29, 479–489 (1984).
  • [13] Nivas, R., Prasad, C. S.: On a structure defined by a tensor field $f(\neq 0)$ of type $(1,1)$ satisfying $f^{5}-a^{2}f=0$. Nep. Math. Sc. Rep. 10(1), 25–30 (1985).
  • [14] Salimov, A. A.: Tensor Operators and Their applications. Nova Science Publ., New York (2013).
  • [15] Salimov, A. A., Çayır, H.: Some Notes On Almost Paracontact Structures. Comptes Rendus de l’Acedemie Bulgare Des Sciences. 66(3), 331-338 (2013).
  • [16] Singh, A.: On $CR-$structures $F-$structures satisfying $ F^{2K+P}+F^{P}=0$. Int. J. Contemp. Math. Sciences. 4, 1029–1035 (2009).
  • [17] Singh, A., Pandey, R. K., Khare, S.: {On horizontal and complete lifts of $(1,1)$ tensor fields $F$ satisfying the structure equation $F(2K+S,S)=0$. International Journal of Mathematics and Soft Computing. 6(1), 143–152 (2016).
  • [18] Yano, K., Patterson, E. M.: Horizontal lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan. 19, 185–198 (1967).
  • [19] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker Inc., New York (1973).
  • [20] Yano, K., Ishihara, S.: On integrabilitiy of a structure f satisfying $f^{3}+f=0$. Quart, J. Math. 25, 217–222 (1964).
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Lovejoy Das 0000-0002-2709-5113

Haşim Çayır

Publication Date January 30, 2020
Acceptance Date September 7, 2019
Published in Issue Year 2020 Volume: 13 Issue: 1

Cite

APA Das, L., & Çayır, H. (2020). On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle. International Electronic Journal of Geometry, 13(1), 94-106. https://doi.org/10.36890/iejg.559746
AMA Das L, Çayır H. On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle. Int. Electron. J. Geom. January 2020;13(1):94-106. doi:10.36890/iejg.559746
Chicago Das, Lovejoy, and Haşim Çayır. “On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle”. International Electronic Journal of Geometry 13, no. 1 (January 2020): 94-106. https://doi.org/10.36890/iejg.559746.
EndNote Das L, Çayır H (January 1, 2020) On the Integrability Conditions and Operators of the F(K + 1),(K − 1) − Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle. International Electronic Journal of Geometry 13 1 94–106.
IEEE L. Das and H. Çayır, “On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle”, Int. Electron. J. Geom., vol. 13, no. 1, pp. 94–106, 2020, doi: 10.36890/iejg.559746.
ISNAD Das, Lovejoy - Çayır, Haşim. “On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle”. International Electronic Journal of Geometry 13/1 (January 2020), 94-106. https://doi.org/10.36890/iejg.559746.
JAMA Das L, Çayır H. On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle. Int. Electron. J. Geom. 2020;13:94–106.
MLA Das, Lovejoy and Haşim Çayır. “On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle”. International Electronic Journal of Geometry, vol. 13, no. 1, 2020, pp. 94-106, doi:10.36890/iejg.559746.
Vancouver Das L, Çayır H. On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle. Int. Electron. J. Geom. 2020;13(1):94-106.
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