On the C-projective vector fields on Randers spaces

TL;DR

This paper characterizes C-projective vector fields on Randers spaces using a new invariant, establishing maximum dimension conditions and providing explicit examples like generalized Funk metrics.

Contribution

It introduces a new invariant for C-projective vector fields on Randers spaces and characterizes maximum dimension cases, including explicit examples.

Findings

The invariant ${f ext extXi}$ is preserved under C-projective vector fields.

Maximum dimension of C-projective algebra is $n(n+2)$ for certain Randers spaces.

Generalized Funk metrics attain the maximum C-projective algebra dimension.

Abstract

A characterization of the C-projective vector fields on a Randers spaces is presented in terms of a recently introduced non-Riemannian quantity defined by Z. Shen and denoted by ${\bf\Xi}$; It is proved that the quantity ${\bf\Xi}$ is invariant for C-projective vector fields. Therefore, the dimension of the algebra of the C-projective vector fields on an $n$-dimensional Randers space is at most $n(n+2)$. The generalized Funk metrics on the $n$-dimensional Euclidean unit ball $\mathbb{B}^n(1)$ are shown to be explicit examples of the Randers metrics with a C-projective algebra of maximum dimension $n(n+2)$. Then, it is also proved that an $n$-dimensional Randers space has a C-projective algebra of maximum dimension $n(n+2)$ if and only if it is locally Minkowskian or (up to re-scaling) locally isometric to the generalized Funk metric. A new projective invariant is also introduced.