On a class of Ricci-flat Douglas metrics (Q2888825)

Let \( \alpha = \sqrt{a_{ij}(x)y^iy^j}\) be a Riemannian metric and \(\beta = b_i(x)y^i\) a \(1\)-form in the coordinates \((x^i,y^i)\) on the tangent bundle of a smooth manifold. The authors consider the \((\alpha , \beta \))-metric \( F = \alpha \phi (\beta / \alpha ) \) on an open set of \(\mathbb{R}^n\) that satisfy (a) \(\beta\) is not parallel with respect to \(\alpha\) , (b) \(F\) is not of Randers type, (c) \(db\neq 0 \) everywhere or \(b = constant\) for \(b(x) =\| \beta_x \| _{\alpha} \). They find conditions for the Ricci curvature \(^{\alpha}Ric\) of \(\alpha\), for the covariant derivative of \(\beta\) and for \(\phi(s), s= \beta / \alpha\) which are necessary and sufficient (taken together) in order that these metrics are Ricci-flat Douglas metrics.