Distinguished dimensions for special Riemannian geometries

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+2nk=3k+2, where k=1,2,4,8k=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 HkHk in dimensions nknk.The necessary and sufficient conditions for the HkHk 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 are the examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections.The torsionless models for the HkHk geometries have the respective symmetry groups G1=SU(3), G2=SU(3)×SU(3), G3=SU(6) and G4=E6. The groups HkHk and GkGk constitute a part of the ‘magic square’ for Lie groups. The ‘magic square’ Lie groups suggest studies of ten other classes of special Riemannian geometries. Apart from the two exceptional cases, they have the structure groups U(3), S(U(3)×U(3)), U(6), E6×SO(2), Sp(3)×SU(2), SU(6)×SU(2), SO(12)×SU(2) and E7×SU(2) and should be considered in respective dimensions 12, 18, 30, 54, 28, 40, 64 and 112. The two ‘exceptional’ cases are: SU(2)×SU(2) geometries in dimension 8 and SO(10)×SO(2) geometries in dimension 32.The case of SU(2)×SU(2) geometry in dimension 8 is examined closer. We determine the tensor that reduces SO(8) to SU(2)×SU(2) leaving the more detailed studies of the geometries based on the magic square ideas to the forthcoming paper.