Sasakian 3-metric as a generalized Miao-Tam equation
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The Miao-Tam equation gives an Einstein metric but the Fischer-Marsden equation does not, when an associated function to these equations is a non-zero constant. In this article, the critical equation $\mathcal{L}_g^\ast (\lambda) = \alpha g$ is introduced that generalizes the famous Miao-Tam equation and the Fischer-Marsden equation and gives an Einstein metric when the associated function is a non-zero constant. It is shown that a Sasakian 3-manifold $(M,g)$ admitting a non-constant solution of the equation $\mathcal{L}_g^\ast (\lambda) = \alpha g$ is Einstein and has constant curvature 1. In addition, if $(M,g)$ is complete, then it is compact Einstein and isometric to a unit sphere. Several consequences are drawn from these results. Finally, an example is constructed to verify the results.
Abstract
The Miao-Tam equation gives an Einstein metric but the Fischer-Marsden equation does not, when an associated function to these equations is a non-zero constant. In this article, the critical equation $\mathcal{L}_g^\ast (\lambda) = \alpha g$ is introduced that generalizes the famous Miao-Tam equation and the Fischer-Marsden equation and gives an Einstein metric when the associated function is a non-zero constant. It is shown that a Sasakian 3-manifold $(M,g)$ admitting a non-constant solution of the equation $\mathcal{L}_g^\ast (\lambda) = \alpha g$ is Einstein and has constant curvature 1. In addition, if $(M,g)$ is complete, then it is compact Einstein and isometric to a unit sphere. Several consequences are drawn from these results. Finally, an example is constructed to verify the results.