The space of associated metrics on a symplectic manifold

Abstract

The spaces of Riemannian metrics on a closed manifold $M$ are studied. On the space $\mathcal{M}$ of all Riemannian metrics on $M$ the various weak Riemannian structures are defined and the corresponding connections are studied. The space $\mathcal{A}M$ of associated metrics on a symplectic manifold $M,\omega$ is considered in more detail. A natural parametrization of the space $\mathcal{A}M$ is defined. It is shown, that $\mathcal{A}M$ is a complex manifold. A curvature of the space $\mathcal{A}M$ and quotient space $\mathcal{A}M/{\mathcal{D}}_{\omega }$ is found. The finite dimensionality of the space of associated metrics of a constant scalar curvature with Hermitian Ricci tensor is shown.

LaTeX, 83 pages