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
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 $T(M^{n})$.
📌 3 Maddede Bu Makale
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Amaç & Hipotez:This paper consists of two main sections.
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Temel Bulgular & Veri:In the first part, we find the integrability conditions of the horizontal lifts of $F((K+1),(K-1))-$ structure satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$.
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Sonuç & Katkı: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 $T(M^{n})$.
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 $T(M^{n})$.
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, Volume: 13 Issue: 1. https://doi.org/10.36890/iejg.559746
Andreou, F. G.: On integrability conditions of a structure $f$ satisfying $f^{5}+f=0$. Tensor N.S. 40, 27–31 (1983).
Çayır, H.: Some Notes on Lifts of Almost Paracontact Structures. American Review of Mathematics and Statistics. 3(1), 52–60 (2015).
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).
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).
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).
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).
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).
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).
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).
Gupta, V.C.: Integrability Conditions of a Structure $F$ Satisfying $F^{K}+F=0$. The Nepali Math. Sc. Report. 14(2), 55-62 (1998).
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry-Volume I. John Wiley & Sons Inc, New York (1963).
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).
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).
Salimov, A. A.: Tensor Operators and Their applications. Nova Science Publ., New York (2013).
Salimov, A. A., Çayır, H.: Some Notes On Almost Paracontact Structures. Comptes Rendus de l’Acedemie Bulgare Des Sciences. 66(3), 331-338 (2013).
Singh, A.: On $CR-$structures $F-$structures satisfying $ F^{2K+P}+F^{P}=0$. Int. J. Contemp. Math. Sciences. 4, 1029–1035 (2009).
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).
Yano, K., Patterson, E. M.: Horizontal lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan. 19, 185–198 (1967).
Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker Inc., New York (1973).
Yano, K., Ishihara, S.: On integrabilitiy of a structure f satisfying $f^{3}+f=0$. Quart, J. Math. 25, 217–222 (1964).