Pontryagin forms on \((4r-2)\)-manifolds and symplectic structures on the spaces of Riemannian metrics
Two theorems are stated in the paper. Let \(M\) be a compact, connected, oriented, smooth \((4k-2)\)-manifold without boundary, and let \(\mathfrak{M}(M)\) be the space of Riemannian metrics on \(M\). In the introduction it is explained that \(\mathfrak{M}(M)\) admits a pre-symplectic structure (closed differential 2-form) that is invariant under the action of \(\mathsf{Diff}^+(M)\). Denote this 2-form by \(\sigma\). The first theorem gives an explicit expression for \(\sigma\). The second theorem states that when \(M\) is a surface of genus greater than 1, then \(\sigma\) naturally corresponds to \(\frac{1}{2\pi^2}\) times the Weil-Petersson symplectic form on the Teichmüller space \(\mathcal{T}(M)\). It is remarked that this second theorem gives a new proof of the fact that the Weil-Petersson metric on \(\mathcal{T}(M)\) is Kähler.
scientific article; zbMATH DE number 6034645
scientific article