A symplectic reduction for pseudo-Riemannian manifolds with compatible almost product structures
type: Item
instance of: scholarly article
title: A symplectic reduction for pseudo-Riemannian manifolds with compatible almost product structures
zbMATH Open document ID: 1084.53070
author: Jerzy J. Konderak
published in: Beiträge zur Algebra und Geometrie
publication date: 2004-09-10T00:00:00Z
full work available at URL: https://eudml.org/doc/232952
full work available at URL: 10.html
review text: This paper is devoted to study the Marsden-Weinstein symplectic reduction procedure of a manifold with additional geometric structures; in particular, a manifold \(M\) endowed with a pseudo-Riemannian metric \(g\) and a compatible almost-product structure \(P\). Then, there is a non-degenerate \(2\)-form \(\omega\) which is compatible with \(P\), and such that, if \(P\) is parallel with respect to the Levi-Civita connection of \(g\), then \(\omega\) is a symplectic form. The author considers the action of a Lie group \(G\) on \(M\) by isometries leaving the almost-product structure \(P\) invariant. Then, assuming a convenient hypothesis, the symplectic reduction procedure is applied, giving a quotient symplectic manifold which is also endowed with a pseudo-Riemannian metric and a compatible almost-product structure. Furthermore, if \(P\) is parallel, then the quotient almost-product structure is parallel too. Finally, as an example, the procedure is applied to reduce the standard pseudo-Riemannian metric on \({\mathbb R}^{2m}\) for obtaining a Fubini-Study type metric on the so-obtained quotient space.
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 2100236
zbMATH Keywords: symplectic manifold
zbMATH Keywords: almost-product structure
zbMATH Keywords: symplectic reduction
zbMATH Keywords: para-Kähler manifold
zbMATH Keywords: Lagrangian foliation
reviewed by: N. Román-Roy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2100236