A classification of certain almost α\alpha-Kenmotsu manifolds

We study $\mathcal D$-homothetic deformations of almost $\alpha$-Kenmotsu structures. We characterize almost contact metric manifolds which are $CR$-integrable almost $\alpha$-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathcal D$-homothetic deformations. If the canonical connection associated to the structure $(\varphi,\xi,\eta,g)$ has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator $h'$ defined by $2\alpha h'=({\mathcal L}_\xi\varphi)\circ\varphi$. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost $\alpha$-Kenmotsu structure. In the case of almost $\alpha$-Kenmotsu $(\kappa,\mu)'$-spaces, this classification gives rise to a scalar invariant depending on the real numbers $\kappa$ and $\alpha$

Odd dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with K\"ahler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triples of abelian almost contact structures, satisfying certain compatibility conditions, which are equivalent to the existence of a sphere of abelian almost contact structures. We obtain the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group $G$ endowed with a left invariant abelian almost 3-contact structure and a compatible left invariant Riemannian metric. We determine conditions for $G$ to admit a special metric connection with totally skew-symmetric torsion, called canonical, which plays the role of the Bismut connection for HKT structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection

Nearly Sasakian geometry and SU(2)SU(2)-structures

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the $5$-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures. By deforming such a $SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Further we prove that both nearly Sasakian and Sasaki-Einstein $5$-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo $SU(2)$-structures which is again strictly related to Sasaki-Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly K\"{a}hler 6-manifold and, in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly K\"{a}hler geometry. In dimension $5$ we determine a connection which parallelizes all the nearly Sasakian $SU(2)$-structure as well as the torsion tensor field. An analogous result holds also for Sasaki-Einstein structures.Comment: We have shortened the introduction and corrected some misprint

The geometry of a 3-quasi-Sasakian manifold

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. In this paper many geometric properties of this class of almost 3-contact metric manifolds are found. In particular, it is proved that the only 3-quasi-Sasakian manifolds of rank 4l+1 are the 3-cosymplectic manifolds and any 3-quasi-Sasakian manifold of maximal rank is necessarily 3-á-Sasakian. Furthermore, the transverse geometry of a 3-quasi-Sasakian manifold is studied, proving that any 3-quasi- Sasakian manifold admits a canonical transversal, projectable quaternionic-K¨ahler structure and a canonical transversal, projectable 3-á-Sasakian structure.CMU

A Note on 3-quasi-Sasakian Geometry

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.Comment: 5 pages, submitted to the Proceedings of the XVI International Fall Workshop on Geometry and Physics, Lisbon, 200

Bi-Legendrian manifolds and paracontact geometry

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.Comment: to appear in Int. J. Geom. Meth. Mod. Phy

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 geometry of a 3-quasi-Sasakian manifold

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. In this paper many geometric properties of this class of almost 3-contact metric manifolds are found. In particular, it is proved that the only 3-quasi-Sasakian manifolds of rank 4l+1 are the 3-cosymplectic manifolds and any 3-quasi-Sasakian manifold of maximal rank is necessarily 3-á-Sasakian. Furthermore, the transverse geometry of a 3-quasi-Sasakian manifold is studied, proving that any 3-quasi- Sasakian manifold admits a canonical transversal, projectable quaternionic-K¨ahler structure and a canonical transversal, projectable 3-á-Sasakian structure.CMU