22 research outputs found
A classification of certain almost -Kenmotsu manifolds
We study -homothetic deformations of almost -Kenmotsu
structures. We characterize almost contact metric manifolds which are
-integrable almost -Kenmotsu manifolds, through the existence of a
canonical linear connection, invariant under -homothetic
deformations. If the canonical connection associated to the structure
has parallel torsion and curvature, then the local
geometry is completely determined by the dimension of the manifold and the
spectrum of the operator defined by . In particular, the manifold is locally equivalent
to a Lie group endowed with a left invariant almost -Kenmotsu
structure. In the case of almost -Kenmotsu -spaces, this
classification gives rise to a scalar invariant depending on the real numbers
and
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 . 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 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 . 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 endowed with a left
invariant abelian almost 3-contact structure and a compatible left invariant
Riemannian metric. We determine conditions for 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 -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 -dimensional
nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a
contact manifold. Focusing on the -dimensional case, we prove that there
exists a one-to-one correspondence between nearly Sasakian structures and a
special class of nearly hypo -structures. By deforming such a
-structure one obtains in fact a Sasaki-Einstein structure. Further we
prove that both nearly Sasakian and Sasaki-Einstein -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 -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 we determine a connection which parallelizes all the nearly
Sasakian -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