Mathematics – Differential Geometry
Scientific paper
2006-01-02
Mathematics
Differential Geometry
Section 9 has been changed, so that the relation between the 6th tensor defining SU(2)xSU(2) geometry in dimension 8 and isopa
Scientific paper
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 n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups H_k\subset SO(n_k), with H_1=SO(3), H_2=SU(3), H_4=Sp(3) and H_8=F_4. This enables studies of special Riemannian geometries with structure groups H_k in dimensions n_k. The neccessary and sufficient conditions for the H_k 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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