A note on the moment map on compact K\"ahler manifolds

We consider compact K\"ahler manifolds acted on by a connected compact Lie group $K$ of isometries in Hamiltonian fashion. We prove that the squared moment map $\|\mu\|^2$ is constant if and only if the manifold is biholomorphically and $K$-equivariantly isometric to a product of a flag manifold and a compact K\"ahler manifold which is acted on trivially by $K$. The authors do not know whether the compactness of $M$ is essential in the main theorem; more generally it would be interesting to have a similar result for (compact) symplectic manifolds.Comment: 4 page

A Hamiltonian stable minimal Lagrangian submanifold of projective space with non-parallel second fundamental form

In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.Comment: 7 page

Actions of vanishing homogeneity rank on quaternionic-Kaehler projective spaces

We classify isometric actions of compact Lie groups on quaternionic-K\"ahler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.Comment: 18 pages. The present version corrects and improves the previous version of the paper entitled "3-coisotropic actions on positive quaternionic-Kaehler manifolds". A key example has been adde

A splitting result for compact symplectic manifolds

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu||^2$ is constant if and only if $K$ is semisimple and the manifold is $K$-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by $K$. In the almost-K\"ahler setting the symplectomorphism turns out to be an isometry.Comment: 5 pages, no figure

Homogeneous Hypercomplex Structures and the Joyce's Construction

We prove that any invariant hypercomplex structure on a homogeneous space $M = G/L$ where $G$ is a compact Lie group is obtained via the Joyce's construction, provided that there exists a hyper-Hermitian naturally reductive invariant metric on $M$.Comment: 11 page

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.Comment: 13 page