Pontryagin forms on \((4r-2)\)-manifolds and symplectic structures on the spaces of Riemannian metrics
type: Item
instance of: scholarly article
title: Pontryagin forms on \((4r-2)\)-manifolds and symplectic structures on the spaces of Riemannian metrics
zbMATH Open document ID: 1250.53075
published in: Differential Geometry and its Applications
publication date: 2012-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0507076
review text: 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.
reviewed by: Sean Lawton
Mathematics Subject Classification ID: 53D30
Mathematics Subject Classification ID: 58D17
zbMATH DE Number: 6034645
zbMATH Keywords: bundle of Riemannian metrics
zbMATH Keywords: diffeomorphism invariance
zbMATH Keywords: equivariant characteristic classes
zbMATH Keywords: universal Pontryagin forms
zbMATH Keywords: pre-symplectic structure
zbMATH Keywords: Weil-Petersson symplectic form
Wikidata QID: Q115356726
author: R. Ferreiro Pérez
author: Jaime Muñoz Masqué
MaRDI profile type: Publication
OpenAlex ID: W2011883581
arXiv ID: math/0507076
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DOI: 10.1016/J.DIFGEO.2012.03.005
description: scientific article; zbMATH DE number 6034645