Randers spaces of constant curvature (Q810885)

Randers spaces were introduced by \textit{R. S. Ingarden} [Trav. Soc. Sci. Lett., Ser. B 45, 60 p. (1957; Zbl 0080.210)]. These are Finsler spaces with metric function \(L(x,y)=\alpha +\beta\), \(\alpha^ 2=a_{ij}(x)y^ iy^ j\) being a Riemannian metric and \(\beta =b_ i(x)y^ i\) a non- zero differential 1-form. A Randers space is of constant curvature K if its Berwald curvature tensor has the form \(H^ i_{hkj}=K(g_{hk}\delta^ i_ j-g_{hj}\delta^ i_ k).\) The calculation of the tensor \(H^ i_{hkj}\) although straightforward is very complicated. Applying an ingenue method, the author determines the necessary and sufficient conditions for a Randers space to be of constant curvature K. These allow to formulate the following classification theorem: A Randers space of dimension more than two is of constant curvature K if and only if it is a ``Randers constant gradient'' or a ``Randers constant translation'' - with some special properties.