NOTES ON CRITICAL ALMOST HERMITIAN STRUCTURES

We discuss the critical points of the functional F(lambda,mu)(J, g) = integral(M)(lambda tau+mu tau*)dv(g) on the spaces of all almost Hermitian structures AH(M) with (lambda, mu) is an element of R(2) - (0, 0), where tau and tau* being the scalar curvature and the *-scalar curvature of (J, g), respectively. We shall give several characterizations of Kahler structure for some special classes of almost Hermitian manifolds, in terms of the critical points of the functionals F(lambda,mu)(J, g) on AH(M). Further, we provide the almost Hermitian analogy of the Hilbert's result.