Irreducible SO(3) geometry in dimension five

Abstract

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $Υ$ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold $(M,g,Υ)$ with Riemannian metric $g$ and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using so(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In particular, it is shown that the integrable models (those with vanishing torsion) are isometric to the symmetric spaces $M_+= SU(3)/SO(3)$, $M_-=SL(3,R)/SO(3)$, $M_0=R^5$. We also find all nearly integrable SO(3) structures with transitive symmetry groups of dimension $d>5$ and some examples for which $d=5$. Given an SO(3) structure $(M,g,Υ)$, we define its "twistor space" T to be the $S^2$-bundle of those unit 2-forms on $M$ which span $R^3=so(3)$. The 7-dimensional twistor manifold T is then naturally equipped with several CR and $G_2$ structures. The ensuing integrability conditions are discussed and interpreted in terms of the Gray-Hervella type classification.