Special Riemannian geometries modeled on distinguished symmetric spaces
  We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)$, $G_3=SU(6)$ 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), $E_6\times SO(2)$, $Sp(3)\times SU(2)$, $SU(6)\times SU(2)$, $SO(12)\times SU(2)$ and $E_7\times SU(2)$. The respective dimensions are: 18, 30, 54, 28, 40, 64 and 112. This list is supplemented by the two `exceptional' cases of $SU(2)\times SU(2)$ geometries in dimension 8 and $SO(10)\times SO(2)$ geometries in dimension 32. 
Pawel Nurowski