Nearly Sasakian geometry and $SU(2)$-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 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.
Cite
@article{arxiv.1410.0942,
title = {Nearly Sasakian geometry and $SU(2)$-structures},
author = {Beniamino Cappelletti-Montano and Giulia Dileo},
journal= {arXiv preprint arXiv:1410.0942},
year = {2022}
}
Comments
We have shortened the introduction and corrected some misprints