Analyzing non-degenerate 2-forms with Riemannian metrics (Q1857372)

A Riemannian metric \(g\) is compatible with a nondegenerate \(2\)-form \(\sigma\) on a \(2m\)-dimensional manifold \(M\) if at each point \(p\in M\) there exists a chart in which \(g(p)\) and \(\sigma(p)\) are the standard Euclidean metric and symplectic form in \(\mathbb R^{2n}\), respectively. In this paper, the author gives variational proofs of several well known results. It is shown that if \((M,\sigma,g)\) is an almost Kähler manifold, then the symplectic form \(\sigma\) is harmonic with respect to the Riemannian metric \(g\). Also, it is proven that every nondegenerate parallel \(2\)-form on a connected manifold \(M\) is Kähler with respect to certain metric on \(M\). Finally, using the Weitzenböck formula and the first property, the author proves that on a locally conformally flat almost Kähler manifold \((M,\sigma,g)\) the non-positive scalar curvature vanishes if and only if the manifold is Kähler. In the second part of the paper the author provides some variations and related results, especially in dimension \(4\).