On the Projective Algebra of Randers Metrics of Constant Flag Curvature

TL;DR

This paper investigates the structure and dimension of the projective algebra of Randers metrics with constant flag curvature, revealing bounds on its size and characterizing its subalgebras.

Contribution

It characterizes the projective algebra of Randers metrics of constant flag curvature as a Lie subalgebra and establishes bounds on its dimension for compact manifolds.

Findings

Dimension of the projective algebra is either n(n+2) or at most n(n+1)/2.

The projective algebra is a Lie subalgebra of that of the underlying Riemannian metric.

Subgroups and invariants of the projective group are studied.

Abstract

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=\alpha+\beta)$ of a Randers space is characterized as a certain Lie subalgebra of the projective algebra $p(M,\alpha)$. Certain subgroups of the projective group $P(M,F)$ and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact $n$-manifold either equals $n(n+2)$ or at most is $\frac{n(n+1)}{2}$.