Four-dimensional naturally reductive homogeneous spaces.(English)Zbl 0631.53039

Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 223-232 (1983).
[For the entire collection see Zbl 0624.00013.]
The authors prove that every simply connected four-dimensional naturally reductive (Riemannian homogeneous) space is either symmetric or decomposable as direct product. This contrasts to the 3-dimensional case, where three types of non-symmetric irreducible Riemannian spaces occur (the Lie group SU(2), the universal covering group of SL(2,$${\mathbb{R}})$$ and the Heisenberg group $$H_ 3$$, all with special left-invariant metrics).

MSC:

 53C30 Differential geometry of homogeneous manifolds

Zbl 0624.00013