The space of associated metrics on a symplectic manifold

2001/08/16 by N. K. Smolentsev, Smolentsev, N. K.
Mathematics · Physics and Astronomy · #58D17 (Primary) 53C15 (Secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C15 #msc:58D17

paper · pdf · doi:10.48550/arxiv.math/0108110

LaTeX, 83 pages

arxiv created 2001/08/16 · openalex publication_date 2001/08/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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