A splitting result for compact symplectic manifolds. (Q2581106)

The setting of the main result of the paper under review is provided by any compact symplectic manifold \((M,\omega)\) which is effectively acted on by a compact Lie group \(K\) in a Hamiltonian fashion. Denote by \(\mu\colon M\to{\mathfrak k}^*\) the corresponding moment map, and assume that the Lie algebra \({\mathfrak k}\) of \(K\) is equipped with an \(\text{Ad}(K)\)-invariant scalar product. This scalar product defines a dual scalar product (hence a norm) on \({\mathfrak k}^*\), and then the function \(\|\mu(\cdot)\|^2\) is constant on \(M\) if and only if the group \(K\) is semisimple and the manifold \(M\) is \(K\)-equivariantly symplectomorphic to the product of a flag manifold and a compact manifold which is trivially acted on by \(K\). In addition, if the symplectic manifold \((M,\omega)\) is equipped with a \(K\)-invariant almost complex structure compatible with \(\omega\), then the aforementioned symplectomorphism can be chosen isometric with respect to the corresponding Riemannian metric on \(M\). This nice result is a symplectic version of a theorem of \textit{A.~Gori} and \textit{F.~Podestà} [Ann. Global Anal. Geom. 26, No.3, 315--318 (2004; Zbl 1084.53068)] on Hamiltonian actions by isometries of a compact Kähler manifold.