On weakly Einstein almost contact manifolds

TL;DR

This paper investigates weakly Einstein metrics on almost contact manifolds, providing bounds on scalar curvature, classifying certain contact metric manifolds, and describing their local geometric structures.

Contribution

It offers new bounds on scalar curvature for Sasakian manifolds, classifies weakly Einstein contact metric (,)-manifolds, and characterizes their local isomorphism types.

Findings

Scalar curvature bounds for Sasakian manifolds with weakly Einstein metrics.

Classification of weakly Einstein contact metric (,)-manifolds with <1.

Identification of local geometric structures for <0 cases.

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.