Nearly Sasakian geometry and $SU(2)$-structures
| Autor: | Cappelletti-Montano, Beniamino, Dileo, Giulia |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Ann. Mat. Pura Appl. 195 (2016), no. 3, 897-922 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10231-015-0496-9 |
| Popis: | 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 misprints |
| Databáze: | arXiv |
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