Tangent bundles endowed with quarter-symmetric non-metric \(\xi\)-connection on 3-dimensional quasi-Sasakian manifolds (Q6619039)

Let \(M\) be an \(n\)-dimensional differentiable manifold of class \(\mathcal{C}^{\infty}\) wit te Levi-Civita connection \(\nabla\). Let \(\chi_1\) and \(\chi_2\) be vector fields on \(M\) then a linear connection \(\tilde{\nabla}\) is known as symmetric connection on \(M\) if the torsion tensor \(T\) of \(\tilde{\nabla}\) defined by \begin{eqnarray*} T(\chi_1, \chi_2) = \tilde{\nabla}_{\chi_1} \chi_2 - \tilde{\nabla}_{\chi_2} \chi_2 - [\chi_1, \chi_2], \end{eqnarray*} is zero, or else it is non-symmetric. If the torsion tensor \(T\) satisfies \begin{eqnarray*} T(\chi_1, \chi_2) = \pi(\chi_2)\chi_1 - \pi(\chi_1)\chi_2, \end{eqnarray*} where \begin{eqnarray*} \pi(\chi_1) = (\xi, \chi_1), \forall \chi \in \mathfrak{F}^{1}_{0}(M), \end{eqnarray*} \(\pi\) is \(1\)-form, \(\phi\) is a tensor field of type \((1,1)\) and \(g\) is the Riemannian metric, then \(\tilde{\nabla}\) is called a quarter-symmetric connection. If the Riemannian metric \(g\) satisfies \(\tilde{\nabla} g \neq 0\), ten \(\tilde{\nabla}\) is said to be a quarter-symmetric non-metric connection. Otherwise if \(\tilde{\nabla} g = 0\), then \(\tilde{\nabla}\) is said to be a quarter-symmetric metric connection. \N\NLet \(M\) be an almost contact metric manifold \((\dim M = 2n+1)\) with \((\phi, \xi, \eta, g)\)-structure with the fundamental \(2\)-form \(\Phi\). Then \(M\) is said to be quasi-Sasakian if the almost contact structure \((\phi, \xi, \eta, g)\) is normal and the fundamental \(2\)-form is closed.\N\NThe authors investigate the tangent bundle of \(3\)-dimensional quasi-Sasakian manifolds endowed with a quarter-symmetric non-metric \(\xi\)-connection. They obtain a theorem on the complete lifts of quarter-symmetric non-metric connections on \(3\)-dimensional quasi-Sasakian manifolds. They also give a theorem on the complete lift of the curvature tensor and Ricci tensor of \(3\)-dimensional quasi-Sasakian manifolds admitting a quarter-symmetric non-metric connection.\N\NFor the entire collection see [Zbl 1537.53001].