Nearly Sasakian geometry and $$SU(2)$$ S U ( 2 ) -structures

Abstract

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 and 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 an 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 Kähler 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 Kähler 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.

On nearly Sasakian and nearly cosymplectic manifolds

“…Recently, a systematic study of nearly Sasakian and nearly cosymplectic manifolds was carried forward in [7]. In that paper, the authors proved that any nearly Sasakian manifold is a contact manifold.…”

Nearly Sasakian manifolds revisited

“…Concerning other dimensions, there have been many attempts of finding explicit examples of proper nearly Sasakian non-Sasakian manifolds until the recent result obtained in [4] showing that every nearly Sasakian structure of dimension greater than five is always Sasakian. Such result depends on the early work [3] by the first and third authors, which in turn draws many properties proved in [6]. This makes the proof to be spread over several different texts with different notation.…”

“…Nevertheless, in recent years several differences between nearly Sasakian and nearly Kähler geometry were pointed out. In particular, in [3] it was proved that the 1-form η of any nearly Sasakian manifold is necessarily a contact form, while the fundamental 2-form of a nearly Kähler manifold is never symplectic unless the manifold is Kähler. A peculiarity of nearly Sasakian five dimensional manifolds, which are not Sasakian, is that upon rescaling the metric one can define a Sasaki-Einstein structure on them.…”

Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds

“…In fact, any normal nearly Sasakian manifold is Sasakian (see [3] and references therein for more details). From then, many papers have appeared on these manifolds and their submanifolds [2], [6], [1] and [19]. In these papers, the geometry is restricted to a Riemannian case and thus, little or no attempt has been made to investigate their lightlike (null) cases.…”