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## On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle

#### Lovejoy DAS [1] , Haşim ÇAYIR [2]

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 $T(M^{n})$.
Integrability conditions, Tachibana operators, $CR-$Submanifolds, $CR-$Stucture, tangent bundle, cotangent bundle
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Primary Language en Mathematics Research Article Orcid: 0000-0002-2709-5113Author: Lovejoy DAS (Primary Author)Institution: Kent State UniversityCountry: United States Author: Haşim ÇAYIR Institution: Giresun ÜniversitesiCountry: Turkey Publication Date : January 30, 2020
 Bibtex @research article { iejg559746, journal = {International Electronic Journal of Geometry}, issn = {}, eissn = {1307-5624}, address = {}, publisher = {Kazım İLARSLAN}, year = {2020}, volume = {13}, pages = {94 - 106}, doi = {10.36890/iejg.559746}, title = {On the Integrability Conditions and Operators of the \$F((K+1),(K-1))-\$ Structure Satisfying \$F\^\{K+1\}+F\^\{K-1\}=0,\$ \$(F\\neq 0,\$ \$K\\eqslantgtr 2)\$ on Cotangent Bundle and Tangent Bundle}, key = {cite}, author = {DAS, Lovejoy and ÇAYIR, Haşim} } APA DAS, L , ÇAYIR, H . (2020). On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle. International Electronic Journal of Geometry , 13 (1) , 94-106 . DOI: 10.36890/iejg.559746 MLA DAS, L , ÇAYIR, H . "On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle". International Electronic Journal of Geometry 13 (2020 ): 94-106 Chicago DAS, L , ÇAYIR, H . "On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle". International Electronic Journal of Geometry 13 (2020 ): 94-106 RIS TY - JOUR T1 - On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle AU - Lovejoy DAS , Haşim ÇAYIR Y1 - 2020 PY - 2020 N1 - doi: 10.36890/iejg.559746 DO - 10.36890/iejg.559746 T2 - International Electronic Journal of Geometry JF - Journal JO - JOR SP - 94 EP - 106 VL - 13 IS - 1 SN - -1307-5624 M3 - doi: 10.36890/iejg.559746 UR - https://doi.org/10.36890/iejg.559746 Y2 - 2019 ER - EndNote %0 International Electronic Journal of Geometry On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle %A Lovejoy DAS , Haşim ÇAYIR %T On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle %D 2020 %J International Electronic Journal of Geometry %P -1307-5624 %V 13 %N 1 %R doi: 10.36890/iejg.559746 %U 10.36890/iejg.559746 ISNAD DAS, Lovejoy , ÇAYIR, Haşim . "On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle". International Electronic Journal of Geometry 13 / 1 (January 2020): 94-106 . https://doi.org/10.36890/iejg.559746 AMA DAS L , ÇAYIR H . On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle. Int. Electron. J. Geom.. 2020; 13(1): 94-106. Vancouver DAS L , ÇAYIR H . On the Integrability Conditions and Operators of the $F((K+1),(K-1))-$ Structure Satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$ on Cotangent Bundle and Tangent Bundle. International Electronic Journal of Geometry. 2020; 13(1): 106-94.

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