Computer algebra and two and three dimensional finsler geometry

Abstract

After a review of 2- and 3-dimensional Finsler spaces from the Berwald and Cartan connections points-of-view, several tensors are explicitly worked out by means of Computer Algebra, and Moór frames are also used to obtain information about the almost flat metric ds2 = dr2 + r2dω2 - dt2 + εdωdt, derived from the Rutz's 2 parameter family of metrics, an non-Riemannian first-order perturbation of the Schwarzschild solution (to Eintein's field equations), which solves, up to order ε, a generalized field equation consisting of a trace-free deviation tensor, Bii = 0. It is determined that such spaces are not Landsberg, are of zero curvature and yet are not projectively flat, all results being up to order ε. Also, only one of the three Cartan's curvature tensor turns out to be null, namely Sijkl = 0(ε2), and the link to Brickell's theorem is mentioned. All calculations are performed using the FINSLER package, based on MAPLE.