Generalized Goldberg Formula

In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally Kaehler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra.Comment: 12 pages, accepted for publication in the Canadian Mathematical Bulleti

Reduction of pre-Hamiltonian actions

We prove a reduction theorem for the tangent bundle of a Poisson manifold $(M, \pi)$ endowed with a pre-Hamiltonian action of a Poisson Lie group $(G, \pi_G)$. In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of $M$. If the manifold $M$ is symplectic and simply connected, the reduced tangent bundle is integrable and its integral symplectic groupoid is the Marsden-Weinstein reduction of the pair groupoid $M \times \bar{M}$.Comment: 18 pages, final version, to appear in Journal of Geometry and Physic

Curvature properties of 3-quasi-Sasakian manifolds

We find some curvature properties of 3-quasi-Sasakian manifolds which are similar to some well-known identities holding in the Sasakian case. As an application, we prove that any 3-quasi-Sasakian manifold of constant horizontal sectional curvature is necessarily either 3-\alpha-Sasakian or 3-cosymplectic.Comment: 7 pages, to appear in Int. J. Geom. Methods Mod. Phys. (IJGMMP

Some remarks on cosymplectic 3-structures

In this note we briefly review some recent results of the authors on the topological and geometrical properties of 3-cosymplectic manifolds.Comment: 6 page

Cosymplectic p-spheres

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 $M \times \mathbb{R}$. 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.Comment: 17 pages, accepted for publication in Journal of Geometry and Physic

Hard Lefschetz Theorem for Sasakian manifolds

We prove that on a compact Sasakian manifold $(M, \eta, g)$ of dimension $2n+1$, for any $0 \le p \le n$ the wedge product with $\eta \wedge (d\eta)^p$ defines an isomorphism between the spaces of harmonic forms $\Omega^{n-p}_\Delta (M)$ and $\Omega^{n+p+1}_\Delta (M)$. Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p}(M)$ and $H^{n+p+1}(M)$. Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.Comment: 19 pages, 1 figure, accepted for publication in the Journal of Differential Geometr

Interests Diffusion in Social Networks

Understanding cultural phenomena on Social Networks (SNs) and exploiting the implicit knowledge about their members is attracting the interest of different research communities both from the academic and the business side. The community of complexity science is devoting significant efforts to define laws, models, and theories, which, based on acquired knowledge, are able to predict future observations (e.g. success of a product). In the mean time, the semantic web community aims at engineering a new generation of advanced services by defining constructs, models and methods, adding a semantic layer to SNs. In this context, a leapfrog is expected to come from a hybrid approach merging the disciplines above. Along this line, this work focuses on the propagation of individual interests in social networks. The proposed framework consists of the following main components: a method to gather information about the members of the social networks; methods to perform some semantic analysis of the Domain of Interest; a procedure to infer members' interests; and an interests evolution theory to predict how the interests propagate in the network. As a result, one achieves an analytic tool to measure individual features, such as members' susceptibilities and authorities. Although the approach applies to any type of social network, here it is has been tested against the computer science research community. The DBLP (Digital Bibliography and Library Project) database has been elected as test-case since it provides the most comprehensive list of scientific production in this field.Comment: 30 pages 13 figs 4 table

A variational formulation of analytical mechanics in an affine space

Variational formulations of statics and dynamics of mechanical systems controlled by external forces are presented as examples of variational principles.Comment: 17 pages, corrected typos, accepted for publication in Rep. Math. Phy

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only it is flat.Comment: 14 pages, LaTeX; some minor misprints correcte