2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62027The 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 ${\mathcal AM}$ of associated metrics on a symplectic manifold $M,ω$ is considered in more detail. A natural parametrization of the space ${\mathcal AM}$ is defined. It is shown, that ${\mathcal AM}$ is a complex manifold. A curvature of the space ${\mathcal AM}$ and quotient space ${\mathcal AM}/{\mathcal D}_ω$ is found. The finite dimensionality of the space of associated metrics of a constant scalar curvature with Hermitian Ricci tensor is shown.LaTeX, 83 pagesDifferential Geometry58D17 (Primary) 53C15 (Secondary)text