Characterizing Finsler spaces which are pseudo-Riemannian of constant curvature

Abstract

Let M be an indefinite Finsler space. The bisector of two points of M is the set of points equidistant from these two points. A bisector is called flat if with any pair of points it contains the extremals joining this pair. In this paper it is shown that M is pseudo-Riemannian of constant curvature if and only if M locally has flat bisectors. Another result is that M is pseudo-Riemannian of constant curvature if and only if M can be reflected locally in each nonnull extremal.