Advanced Search

We consider generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds. First, we prove that a complete Sasakian manifold M admitting a generalized m-quasi-Einstein metric is compact and isometric to the unit sphere $S^{2n+1}$. Next, we generalize this to complete K-contact manifolds with m ≠ 1.

We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g,\pm \omega )$ with constant scalar curvature is either Einstein, or the dual field of $\omega$ is Killing. Next, let $(M^n,g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g,\pm \omega )$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

We study certain contact metrics satisfying the Miao-Tam critical condition. First, we prove that a complete K-contact metric satisfying the Miao-Tam critical condition is isometric to the unit sphere $S^{2n+1}$. Next, we study (κ,μ)-contact metrics satisfying the Miao-Tam critical condition.