Special Riemannian geometries modeled on distinguished symmetric spaces

Pawel Nurowski

paper

Special Riemannian geometries modeled on distinguished symmetric spaces

arXiv:math/0603663

Abstract

We propose studies of special Riemannian geometries with structure groups $H_{1} = S O (3) \subset S O (5)$ , $H_{2} = S U (3) \subset S O (8)$ , $H_{3} = S p (3) \subset S O (14)$ and $H_{4} = F_{4} \subset S O (26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_{1} = S U (3)$ , $G_{2} = S U (3) \times S U (3)$ , $G_{3} = S U (6)$ and $G_{4} = E_{6}$ . The groups $H_{k}$ and $G_{k}$ constitute a part of the `magic square' for Lie groups. Apart from the $H_{k}$ geometries in dimensions $n_{k}$ , the `magic square' Lie groups suggest studies of a finite number of other special Riemannian geometries. Among them the smallest dimensional are U(3) geometries in dimension 12. The other structure groups for these Riemannian geometries are: $S (U (3) \times U (3))$ , U(6), $E_{6} \times S O (2)$ , $S p (3) \times S U (2)$ , $S U (6) \times S U (2)$ , $S O (12) \times S U (2)$ and $E_{7} \times S U (2)$ . The respective dimensions are: 18, 30, 54, 28, 40, 64 and 112. This list is supplemented by the two `exceptional' cases of $S U (2) \times S U (2)$ geometries in dimension 8 and $S O (10) \times S O (2)$ geometries in dimension 32.

A written version of a talk at the workshop `Special geometries in mathematical physics' held in Kühlungsborn, March 2006

References in corpus (1)

On types of non-integrable geometries