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).


53C30 Differential geometry of homogeneous manifolds


Zbl 0624.00013