Beniamino Cappelletti-Montano

A survey on cosymplectic geometry

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKahler manifolds. We also mention some of their applications to time-dependent mechanics.

Examples of compact K-contact manifolds with no Sasakian metric

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.

Sasaki–Einstein and paraSasaki–Einstein metrics from (κ,μ)-structures

We prove that every contact metric (κ,μ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ for which such metrics are Sasaki–Einstein and paraSasaki–Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (κ,μ)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of (κ,μ)-spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.

A non-Sasakian Lefschetz K-contact manifold of Tievsky type

We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of

Cosymplectic p-spheres

K

Hard Lefschetz Theorem for Vaisman manifolds

-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

Hard Lefschetz Theorem for Sasakian manifolds

Abstract We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic p -sphere are studied. To any taut cosymplectic circle on a three-dimensional manifold M we are able to canonically associate a complex structure and a conformal symplectic couple on M × R . We prove that a cosymplectic circle in dimension three is round if and only if it is taut. On the other hand, we provide examples in higher dimensions of cosymplectic circles which are taut but not round and examples of cosymplectic circles which are round but not taut.

Nearly Sasakian geometry and SU(2)-structures

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual

Almost formality of quasi-Sasakian and Vaisman manifolds with applications to nilmanifolds

1

Nearly Sasakian geometry and

-form coincides with the Lee