63 research outputs found
Bi-paracontact structures and Legendre foliations
We study almost bi-paracontact structures on contact manifolds. We prove that
if an almost bi-paracontact structure is defined on a contact manifold
, then under some natural assumptions of integrability, carries
two transverse bi-Legendrian structures. Conversely, if two transverse
bi-Legendrian structures are defined on a contact manifold, then 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 -nullity distribution canonically carries an
almost bi-paracontact structure and we apply the previous results to the theory
of contact metric -spaces.Comment: To appear on: Kodai Mathematical Journa
Contact metric -spaces as bi-Legendrian manifolds
We regard a contact metric manifold whose Reeb vector field belongs to the
-nullity distribution as a bi-Legendrian manifold and we study
its canonical bi-Legendrian structure. Then we characterize contact metric
-spaces in terms of a canonical connection which can be naturally
defined on them.Comment: To appear on Bull. Austral. Math. So
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
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
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
Hard Lefschetz Theorem for Sasakian manifolds
We prove that on a compact Sasakian manifold of dimension
, for any the wedge product with
defines an isomorphism between the spaces of harmonic forms
and . Therefore it induces
an isomorphism between the de Rham cohomology spaces and
. 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
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 -sphere are studied. To any taut cosymplectic circle on a
three-dimensional manifold we are able to canonically associate a complex
structure and a conformal symplectic couple on . 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
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