Certain types of metrics on almost coKähler manifolds

Abstract

In this paper, we study an almost coKähler manifold admitting certain metrics such as \(*\)-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKähler 3-manifold (M, g) admitting a \(*\)-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKähler \((\kappa ,\mu )\)-almost coKähler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a \((\kappa , \mu )\)-almost coKähler manifold (M, g) is coKähler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKähler manifolds which are non-coKähler.