arXiv subjects
Publications and source records attributed to Beniamino Cappelletti Montano.
The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $ \mathbb{B}^N\times \mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $η$-Einstein case.
Inspired by the analogous result in the algebraic setting (Theorem 1) we show (Theorem 2) that the product $M \times \mathbb{R}P^n$ of a closed and orientable topological manifold $M$ with the $n$-dimensional real projective space cannot be topologically locally flat embedded into $\mathbb{R}P^{m + n + 1}$ for all even $n > m$.
We prove that on a compact Sasakian manifold $(M, η, g)$ of dimension $2n+1$, for any $0 \le p \le n$ the wedge product with $η\wedge (dη)^p$ defines an isomorphism between the spaces of harmonic forms $Ω^{n-p}_Δ(M)$ and $Ω^{n+p+1}_Δ(M)$. 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.
We prove that any non-Sasakian 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 κand μfor 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 topological and geometrical properties of (κ,μ)-spaces related to the existence of Eistein-Weyl and Lorentzian Sasakian Einstein structures.
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-α-Sasakian or 3-cosymplectic.
In this note we briefly review some recent results of the authors on the topological and geometrical properties of 3-cosymplectic manifolds.
We continue the program of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds which are the closest odd-dimensional analogue of hyper-Kaehler structures. We show that there is an action of the Lie algebra so(4,1) on the basic cohomology spaces of a compact 3-cosymplectic manifold with respect to the Reeb foliation. This implies some topological obstructions to the existence of such structures which are expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kaehler factor and an abelian three dimensional Lie group. Nevertheless, we present a nontrivial example of compact 3-cosymplectic manifold which is not the global product of a hyper-Kaehler manifold and a flat 3-torus.
We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,η)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the $(κ,μ)$-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric $(κ,μ)$-spaces.
We prove that any contact metric $(κ,μ)$-space $(M,ξ,ϕ,η,g)$ admits a canonical paracontact metric structure which is compatible with the contact form $η$. We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold $(M,η)$ a sequence of compatible contact and paracontact metric structures verifying nullity conditions. The behavior of that sequence, related to the Boeckx invariant $I_M$ and to the bi-Legendrian structure of $(M,ξ,ϕ,η,g)$, is then studied. Finally we are able to define a canonical Sasakian structure on any contact metric $(κ,μ)$-space whose Boexkx invariant satisfies $|I_M|>1$.
In this paper we study the foliated structure of a contact metric $(κ,μ)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation to the Boeckx's classification of contact metric $(κ,μ)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric $(κ,μ)$-structure. Finally we prove that any contact metric $(κ,μ)$-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.
We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendrian and for paracontact manifolds. Finally new classes of examples of paracontact manifolds are presented.
We regard a contact metric manifold whose Reeb vector field belongs to the $(κ,μ)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(κ,μ)$-spaces in terms of a canonical connection which can be naturally defined on them.
We find necessary and sufficient conditions for the bi-Legendrian connection $\nabla$ associated to a bi-Legendrian structure $(\cal F,\cal G)$ on a contact metric manifold $(M,ϕ,ξ,η,g)$ being a metric connection and then we give conditions ensuring that $\nabla$ coincides with the (generalized) Tanaka-Webster connection of $(M,ϕ,ξ,η,g)$. 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.
3-quasi-Sasakian manifolds were recently studied by the authors as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. In this paper some geometric properties of this class of almost 3-contact metric manifolds are briefly reviewed, with an emphasis on those more related to physical applications.
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.
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.