Abstract
The goal of this note is to investigate some properties of the critical point equations on the $3-$dimensional $f-$ cosymplectic manifolds. We obtain some geometric equations on the $3-$dimensional $f-$ cosymplectic manifolds which admit critical point equations. We give a relation between $f$ and $tilde{f}$ for a CPE metric on the three dimensional $f-$cosymplectic manifold to be Einstein. Also we obtain an eigenvalue of the Laplace operator on the $3-$dimensional $f-$ cosymplectic manifolds with CPE metrics.
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