Andrea Spiro

Six-dimensional nearly Kähler manifolds of cohomogeneity one

Abstract We consider six-dimensional strict nearly Kahler manifolds acted on by a compact, cohomogeneity one automorphism group G . We classify the compact manifolds of this class up to G -diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G is simple.

Kähler-Ricci solitons on homogeneous toric bundles

Abstract We deal with homogeneous toric bundles M over generalized flag manifolds GC /P, where G is a compact semisimple Lie group and P a parabolic subgroup. Using symplectic data, we provide a simple characterization of the homogeneous toric bundles M which are Fano; we then show that a homogeneous toric bundle M admits a Kähler-Ricci solitonic metric if and only if it is Fano.

Some topological properties of cohomogeneity one manifolds with negative curvature

This paper is aimed at studying negatively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. The orbit space of such a manifold M is proved to be always homeomorphic to ℝ or ℝ+ and this second case may occur only when either the singular orbit is a geodesic of M or when the space is simply connected. Several corollaries are given.

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.

Compact homogeneous CR manifolds

We classify all compact simply connected homogeneous CR manifolds M of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group G(M) of automorphisms of M. We characterize also the standard homogeneous CR manifolds as the homogeneous CR manifolds whose group G(M) in not semisimple.

Kahler manifolds with large isometry group

A Riemannian manifold is called of cohomogeneity one if it is acted on by a closed Lie group G of isometrics with principal orbits of codimension one. This class of manifolds have at least two good reasons for being considered particularly appealing: their degree of symmetry is so high that classification theorems of purely algebraic nature are still possible in several situations (see e.g. [5], [1], [2], [10]); at the same time, they allow to construct non homogeneous examples of Riemannian manifolds with special geometric properties, like Einstein metrics, exceptional holonomy (see e.g. [4]). We are interested in compact non homogeneous Kahler-Einstein manifolds and several of them have been already constructed by Koiso and Sakane in the cohomogeneity one category (see e.g. [15], [13]). The aims of the present paper consist in giving an explicit classification of compact cohomogeneity one Kahler manifolds with vanishing first Betti number and in using it for obtaining a complete list of the Kahler-Einstein manifolds in that family. It is well known (see e.g. [9]) that the vanishing of &ι(M) implies the existence of a G-equivariant moment mapping μ: M —> g* and this fact has an important consequence on the algebraic structure of G. In fact, we prove that (see Lemma 2.2) if G is semisimple and G/L = G(p) is a regular orbit on M, with Q and [ Lie algebras of G and L respectively, then the centralizer G0(l) is of the form

A remark on locally homogeneous Riemannian spaces

A locally homogeneous Riemannian space is called non-regular if it is not locally isometric to any globally homogeneous Riemannian space. We show that no non-regular space has non positive Ricci tensor and that a theorem by Alkseevski-Kimelfeld may be extended to the class of locally homogeneous spaces: i.e. any locally homogeneous Riemannian space with zero Ricci tensor is locally euclidean.

Cohomogeneity one manifolds and hypersurfaces of the euclidean space

This paper is aimed at studying compact hypersurfaces of the euclidean space which are supposed to be Riemannian manifolds of cohomogeneity one, namely acted on by a Lie group of intrinsic isometries with principal orbits of codimension one. We give necessary and sufficient conditions on the structure of the Riemannian metric in order that a hypersurface of such kind turns out to be a revolution hypersurface.

Eversive maps of bounded convex domains in ℂ n+1

A duality principle, relating the geometry of the Kobayashi metric with the CR geometry of the boundaries of smoothly bounded, strongly convex domains in ℂn+1 is established. A characterization of the holomorphic Jacobi vector fields of those domains is also given.

GROUPS ACTING TRANSITIVELY ON COMPACT CR MANIFOLDS OF HYPERSURFACE TYPE

Let M = G=L be a compact homogeneous manifold with G acting eectively and with a G-invariant CR structure of hypersurface type; then any maximal compact subgroup K G acts transitively on M.