Nullity conditions in paracontact geometry

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $\tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(\kappa,\mu)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $\mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.Comment: Different. Geom. Appl. (to appear

A comparative analysis on serious adverse events reported for COVID-19 vaccines in adolescents and young adults

This study aims to assess the safety profile of COVID-19 vaccines (mRNA and viral vector vaccines) in teenagers and young adults, as compared to Influenza and HPV vaccines, and to early data from Monkeypox vaccination in United States. Methods: We downloaded data from the Vaccine Adverse Event Reporting System (VAERS) and collected the following Serious Adverse Events (SAEs) reported for COVID-19, Influenza, HPV and Monkeypox vaccines: deaths, life-threatening illnesses, disabilities, hospitalizations. We restricted our analysis to the age groups 12–17 and 18–49, and to the periods December 2020 to July 2022 for COVID-19 vaccines, 2010–2019 for Influenza vaccines, 2006–2019 for HPV vaccines, June 1, 2022 to November 15, 2022 for Monkeypox vaccine. Rates were calculated in each age and sex group, based on an estimation of the number of administered doses. Results: Among adolescents the total number of reported SAEs per million doses for, respectively, COVID-19, Influenza and HPV vaccines were 60.73, 2.96, 14.62. Among young adults the reported SAEs rates for, respectively, COVID-19, Influenza, Monkeypox vaccines were 101.91, 5.35, 11.14. Overall, the rates of reported SAEs were significantly higher for COVID-19, resulting in a rate 19.60-fold higher than Influenza vaccines (95% C.I. 18.80–20.44), 4.15-fold higher than HPV vaccines (95% C.I. 3.91–4.41) and 7.89-fold higher than Monkeypox vaccine (95% C.I. 3.95–15.78). Similar trends were observed in teenagers and young adults with higher Relative Risks for male adolescents. Conclusion: The study identified a risk of SAEs following COVID-19 vaccination which was markedly higher compared to Influenza vaccination and substantially higher compared to HPV vaccination, both for teenagers and young adults, with an increased risk for the male adolescents group. Initial, early data for Monkeypox vaccination point to significantly lower rates of reported SAEs compared to those for COVID-19 vaccines. In conclusion these results stress the need of further studies to explore the bases for the above differences and the importance of accurate harm-benefit analyses, especially for adolescent males, to inform the COVID-19 vaccination campaign

3-quasi-Sasakian manifolds

In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.Comment: 17 pages, minor modifications, references update

The curvature tensor of (\ka,\mu,\nu)-contact metric manifolds

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat

The foliated structure of contact metric (k,μ)-spaces

In this paper we study the foliated structure of a contact metric (k,μ)-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation to the Boeckx's classification of contact metric (k,μ)-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric (k,μ)-structure. Finally we prove that any contact metric (k,μ)-space M whose Boeckx invariant I_M is different from \pm 1 admits a compatible Sasakian or Tanaka-Webster parallel structure according to the circumstance that |I_M|>1 or |I_M|<1, respectively

Bi-Legendrian connections

We define the concept of a bi-Legendrian connection associated to a bi- Legendrian structure on an almost S-manifold M^{2n+r}. Among other things, we compute the torsion of this connection and prove that the curvature vanishes along the leaves of the bi-Legendrian structure. Moreover, we prove that if the bi-Legendrian connection is flat, then the bi-Legendrian structure is locally equivalent to the standard structure on R^{2n+r}

3-structures with torsion

We find conditions which ensure the integrability of the canonical 3-dimensional distribution V spanned by the Reeb vector fields of an almost 3-contact manifold, showing by an explicit counterexample that the normality of the structures does not necessarily imply the integrability of V. Then we focus on those almost 3-contact metric manifolds for which V is integrable and we define an appropriate notion of almost 3-contact metric connection with torsion. The geometry of an almost 3-contact manifold with torsion is then studied and put in relation with the well-known HKT-geometry

A note on Legendre foliations

In a previous paper the author proved that any flat Legendre foliation on a contact manifold, under certain natural assumptions, admits an Ehresmann connection. In this note we extend such result also to the non-flat case

Some remarks on the generalized Tanaka-Webster connection of a contact metric manifold

We find necessary and sufficient conditions for the bi-Legendrian connection \nabla associated to a bi-Legendrian structure (F,G) on a contact metric manifold (M,\phi,\xi,\eta,g) being a metric connection and then we give conditions ensuring that \nabla coincides with the (generalized) Tanaka-Webster connection of (M,\phi,\xi,\eta,g). Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold

Characteristic classes and Ehresmann connections for Legendrian foliations

Abstract. We find a necessary and sufficient condition for the (local) projectabil- ity of a Legendrian foliation of an almost S-manifold onto a Lagrangian foliation of a symplectic manifold. In the context of Legendrian foliations, this result will be used for proving a Darboux theorem and some results about primary and secondary char- acteristic classes. Finally we show that, under suitable assumptions, every Legendrian foliation admits an Ehresmann connection