Calabi Yaj manifolds, SU (3) holonomy group, complex manifolds, seiberg- witten equations, anti self dual forms
Yes ! There are many compact examples. These are the spaces which called Calabi-Yau manifolds. See Berger's list for possible Lie Groups that can occur as holonomy group as a non-symmetric irreducible space: https://en.m.wikipedia.org/wiki/Holonomy#The_Berger_classification
I am not working in this area. I am not aware of the mentioned things.
Yes ! There are many compact examples. These are the spaces which called Calabi-Yau manifolds. See Berger's list for possible Lie Groups that can occur as holonomy group as a non-symmetric irreducible space: https://en.m.wikipedia.org/wiki/Holonomy#The_Berger_classification
Thanks great profesor Mohamed El Naschie...for your helpful explanations..
Nearly Kahler manifolds offer a source of examples. You can read Butruille's paper http://arxiv.org/abs/math/0612655 as a survey. In this paper (a translation from a french paper), he proves that there are four homogeneous nearly Kahler manifolds. There is, for instance, a metric g on the six-sphere for which Hol(g)=SU(3). This metric arises as the quotient G2/SU(3). Up to recently, no non-homogeneous example were known. This year (2015) a paper by Foscolo and Haskins established the existence of the first complete inhomogeneous nearly Kaehler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kaehler structure on the 6-sphere. Their paper is available at http://arxiv.org/abs/1501.07838 .
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