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 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.
Cite
@article{arxiv.math/0601020,
title = {Distinguished dimensions for special Riemannian geometries},
author = {Pawel Nurowski},
journal= {arXiv preprint arXiv:math/0601020},
year = {2007}
}
Comments
Section 9 has been changed, so that the relation between the 6th tensor defining SU(2)xSU(2) geometry in dimension 8 and isoparametric hypersurfaces with 6 different principal curvatures is explicit. A 5-form reducing GL(14,R) to SP(3) is added to the Appendix B