New Quasi-Einstein Metrics on a Two-Sphere

Abstract We construct all axi-symmetric non-gradient m -quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to $$m=2$$ m = 2 , as well as a family of new regular metrics with $$m\ne 2$$ m ≠ 2 given in terms of hypergeometric functions. We also show that in the case $$m=-1$$ m = - 1 with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.