A symplectic reduction for pseudo-Riemannian manifolds with compatible almost product structures
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.
scientific article; zbMATH DE number 2100236