Ricci-flat Douglas (α,β)(α,β)-metrics

In this paper, we study Ricci-flat (α,β)(α,β)-metrics which are defined by a Riemann metric α and a 1-form β   on a C∞C∞ manifold M  . We prove that an (α,β)(α,β)-metric of Randers type is Ricci-flat Douglas metric if and only if it is a Berwald metric and α   is Ricci-flat. Further, we characterize completely Ricci-flat Douglas (α,β)(α,β)-metrics of non-Randers type on M   when the dimension dimM⩾3.