Weakly Einstein almost cosymplectic $3$-manifolds
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초록
In this paper, the first thing we prove is that a weakly Einstein cosymplectic 3-manifold is flat and Einstein. Next, we prove that a strictly almost cosymplectic $3$-manifold $M$ is weakly Einstein if and only if $M$ has the Ricci tensor of rank one. In particular, if $M$ is strictly $H$-almost cosymplectic $3$-manifolds, then it is locally isomorphic to the Minkowski motion group $E_{1,1}$ equipped with a left invariant almost cosymplectic structure with $a^2=b^2$. Moreover, we find that there does not exist a weakly Einstein strictly almost cosymplectic $3$-manifold with $\nabla_{\xi} h=-2\alpha h \varphi$, for any non-zero constant $\alpha$.
Abstract
In this paper, the first thing we prove is that a weakly Einstein cosymplectic 3-manifold is flat and Einstein. Next, we prove that a strictly almost cosymplectic $3$-manifold $M$ is weakly Einstein if and only if $M$ has the Ricci tensor of rank one. In particular, if $M$ is strictly $H$-almost cosymplectic $3$-manifolds, then it is locally isomorphic to the Minkowski motion group $E_{1,1}$ equipped with a left invariant almost cosymplectic structure with $a^2=b^2$. Moreover, we find that there does not exist a weakly Einstein strictly almost cosymplectic $3$-manifold with $\nabla_{\xi} h=-2\alpha h \varphi$, for any non-zero constant $\alpha$.