Some Critical Almost Hermitian Structures

Let M be an even dimensional compact smooth manifold admitting an almost complex structure. Let \({{(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)}\) . We discuss the critical points of the functional \({\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g}\) on the space of all almost Hermitian structures \({\mathcal{AH}(M)}\) on M and its subspace \({{\mathcal{AH}_{c}(M)}}\) with a certain positive constant c, where τ and τ * are the scalar curvature and the *-scalar curvature of (J, g), respectively. Further, we provide some examples illustrating our arguments.