Fabio Podestà

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.

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kahler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

Totally Geodesic Orbits of Isometries

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non-semisimple Lie group with an hypersurface orbit.

Positively curved 7-dimensonal manifolds

We deal with seven dimensional compact Riemannian manifolds of positive curvature which admit a cohomogeneity one action by a compact Lie group G. We prove that the manifold is diffeomorphic to a sphere if the dimension of the semisimple part of G is bigger than 6.

HOMOGENEOUS HERMITIAN MANIFOLDS AND SPECIAL METRICS

We consider non-Kähler compact complex manifolds which are homogeneous under the action of a compact Lie group of biholomorphisms and we investigate the existence of special (invariant) Hermitian metrics on these spaces. We focus on a particular class of such manifolds comprising the case of Calabi–Eckmann manifolds and we prove the existence of an invariant Hermitian metric which is Chern–Einstein, namely whose second Chern–Ricci tensor of the associated Chern connection is a positive multiple of the metric itself. The uniqueness and the property of being astheno-Kähler are also discussed.

On the automorphism group of a symplectic half-flat 6-manifold

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat SU ⁢ ( 3 ) {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by min ⁡ { 5 , b 1 ⁢ ( M ) } {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on T ⁢ 𝕊 3 {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of SO ⁢ ( 4 ) {\mathrm{SO}(4)} .

Homogeneous symplectic half-flat 6-manifolds

We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result of Wolf and Gray allowing one to show the nonexistence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case, we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This property of the Ricci tensor is characterized in the general case.

On the first eigenvalue of invariant Kähler metrics

Given a simply connected compact generalized flag manifold M together with its invariant Kähler–Einstein metric

On the automorphism group of a closed G2-structure

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