On some classes of \(3\)-dimensional generalized \((\kappa ,\mu)\)-contact metric manifolds (Q2813405)

The present paper is concerned with generalized \((\kappa,\mu)\)-contact metric manifolds \((M,\phi,\xi,\eta, g)\). The main results are as follows: A 3-dimensional generalized \((\kappa,\mu)\)-contact metric manifold is locally \(\phi\)-symmetric if and only if \(\kappa\) and \(\mu\) are constants. A \(\xi\)-concirculary flat 3-dimensional non-Sasakian generalized \((\kappa,\mu)\)-contact metric manifold is either flat or reduces to a \((\kappa,\mu)\)-contact metric manifold. A locally \(\phi\)-concircularly symmetric 3-dimensional generalized \((\kappa,\mu)\)-contact metric manifold reduces to a \((\kappa,\mu)\)-contact metric manifold. A 3-dimensional non-Sasakian generalized \((\kappa,\mu)\)-contact metric manifold satisfying the condition \(R. \widetilde{C} = 0\) (\(\widetilde{C}\) being the concircular curvature tensor) is either a \((\kappa,\mu)\)-contact manifold, flat, or of constant \(\xi\)-sectional curvature \(\kappa < 1\) and constant \(\phi\)-sectional curvature \(-1\). A 3-dimensional generalized \((\kappa,\mu)\)-contact metric manifold satisfying the condition \(\widetilde{C}.S = 0\) (\(S\) being the scalar curvature) is either Sasakian, flat, or of constant \(\xi\)-sectional curvature \(\kappa< 1\) and constant \(\phi\)-sectional curvature \(-1\), or a \((\kappa,\mu)\)-contact metric manifold.NEWLINENEWLINENEWLINEFinally, an example of a generalized \((\kappa,\mu)\)-contact metric manifold verifying the first result is given.