Distinguished dimensions for special Riemannian geometries
Abstract
The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions nk=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups Hk⊂ SO(nk), with H1=SO(3), H2=SU(3), H4=Sp(3) and H8=F4. This enables studies of special Riemannian geometries with structure groups Hk in dimensions nk. The neccessary and sufficient conditions for the Hk geometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU(3) geometries in dimension 8 admitting characteristic connection are provided. Among them there are examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections.
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