On THE PROJECTIVE ALGEBRA OF SOME (Α, Β)-Metrics OF ISOTROPIC S-Curvature

Abstract

In this paper, we study projective algebra, p(M, F), of special (α, β)-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We characterize p(M, F) of Matsumoto and square metrics of isotropic S-curvature of dimension n ≥ 3 as a certain Lie sub-algebra of the Killing algebra k(M, α). We also show that F has a maximum projective symmetry if and only if F either is a Riemannian metric of constant sectional curvature or locally Minkowskian.

On a projectively invariant pseudo-distance in Finsler geometry

“…Here we use notations of [7,8]. Without pretending to be exhaustive we quote some more significant works in projective geometry [9][10][11][12][13].…”

“…We call d M (x, y) the pseudo-distance of any two points x and y on M. By means of the property (11) of Schwarzian derivative and the fact that the projective parameter is invariant under fractional transformations, the pseudodistance d M is projectively invariant. Proposition 3.…”