Some notes on lifts of the F((υ+1),λ²(υ-1))-structure on cotangent and tangent bundle

Haşim Çayir

doi:10.31801/cfsuasmas.712861

Abstract

The F^{v+1}-λ²F^{v-1}=0 structure (v≥3) have been studied by Kim J. B. K75 . Later, Srivastava S.K studied on the complete lifts of (1,1) tensor field F satisfying structure F^{v+1}-λ²F^{v-1}=0 and extended in Mⁿ to cotangent bundle. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete and horizontal lifts of F^{v+1}-λ²F^{v-1}=0. Later, we get the results of Tachibana operators applied to vector and covector fields according to the complete and horizontal lifts of F((v+1),λ²(v-1)) -structure and the conditions of almost holomorfic vector fields in cotangent bundle T^{∗}(Mⁿ). Finally, we have studied the purity conditions of Sasakian metric with respect to the lifts of F^{v+1}-λ²F^{v-1}=0-structure. In the second part, all results obtained in the first section were investigated according to the complete and horizontal lifts of the F^{v+1}-λ²F^{v-1}=0 structure in tangent bundle T(Mⁿ).

Highlights

The investigation for the integrability of tensorial structures on manifolds and extension to the tangent or cotangent bundle, whereas the de...ning tensor ...eld satis...es a polynomial identity has been an actively discussed research topic in the last 50 years, initiated by the fundamental works of Kentaro Yano and his collaborators, see for example [25]
Let F be a tensor ...eld of type (1; 1) satisfying F +1 !v; (F +1) 2F 1 = 0 in M n ! T (M n): The Nijenhuis tensor of a (1; 1) tensor ...eld F of M n is given by NF = [F X; F Y ] F [X; F Y ] F [F X; Y ] + F 2 [X; Y ]
We get the results of Tachibana operators applied to vector and covector ...elds according to the horizontal lifts of the structure F +1 rF 1 = 0 in cotangent bundle T (M n)

Summary

Introduction

The investigation for the integrability of tensorial structures on manifolds and extension to the tangent or cotangent bundle, whereas the de...ning tensor ...eld satis...es a polynomial identity has been an actively discussed research topic in the last 50 years, initiated by the fundamental works of Kentaro Yano and his collaborators, see for example [25]. Srivastava S.K studied on the complete lifts of (1; 1) tensor ...eld F satisfying structure F v+1 2F v 1 = 0 and extended in M n to cotangent bundle [21]. We get the results of Tachibana operators applied to vector and covector ...elds according to the complete and horizontal lifts of F +1 2F 1 = 0 structure and the conditions of almost holomor...c vector ...elds in cotangent bundle T (M n). All results obtained in the ...rst section were investigated according to the complete and horizontal lifts of the F ( + 1) ; 2 ( 1) -structure in tangent bundle T (M n). Let M n be a di¤erentiable manifold of class C1 and of dimension n and let T (M n) denote the cotangent bundle of M.

Main Results

LY F 1 X C

H LY v X v