On weakly Einstein almost contact manifolds
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In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a $(2n+1)$-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature $s$ satisfies $-6\leqslant s \leqslant 6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s \leqslant 2n(2n+1)$ for $n\geqslant2$. Secondly, for a $(2n+1)$-dimensional weakly Einstein contact metric $(\kappa,\mu)$-manifold with $\kappa<1$, we prove that it is flat or is locally isomorphic to the Lie group $SU(2)$, $SL(2)$, or $E(1,1)$ for $n=1$ and that for $n\geqslant2$ there are no weakly Einstein metrics on contact metric $(\kappa,\mu)$-manifolds with $0<\kappa<1$. For $\kappa<0$, we get a classification of weakly Einstein contact metric $(\kappa,\mu)$-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic $(\kappa,\mu)$-manifold with $\kappa<0$ is locally isomorphic to a solvable non-nilpotent Lie group.
Abstract
In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a $(2n+1)$-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature $s$ satisfies $-6\leqslant s \leqslant 6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s \leqslant 2n(2n+1)$ for $n\geqslant2$. Secondly, for a $(2n+1)$-dimensional weakly Einstein contact metric $(\kappa,\mu)$-manifold with $\kappa<1$, we prove that it is flat or is locally isomorphic to the Lie group $SU(2)$, $SL(2)$, or $E(1,1)$ for $n=1$ and that for $n\geqslant2$ there are no weakly Einstein metrics on contact metric $(\kappa,\mu)$-manifolds with $0<\kappa<1$. For $\kappa<0$, we get a classification of weakly Einstein contact metric $(\kappa,\mu)$-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic $(\kappa,\mu)$-manifold with $\kappa<0$ is locally isomorphic to a solvable non-nilpotent Lie group.
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