Some cohomogeneity one Einstein–Randers metrics on 4-manifolds

Abstract

The Page metric on [Formula: see text] is a cohomogeneity one Einstein–Riemannian metric, and is the only known cohomogeneity one Einstein–Riemannian metric on compact [Formula: see text]-manifolds. It has been a long standing problem whether there exists another cohomogeneity one Einstein–Riemannian metric on [Formula: see text]-manifolds. In this paper, we construct some examples of cohomogeneity one Einstein–Randers metrics on simply connected 4-manifolds. This shows that, although cohomogeneity one Einstein–Riemmian 4-manifolds are rare, non-Riemannian examples may exist at large.