On the $C$-projective vector fields on Randers spaces
J. Korean Math. Soc. 2020 Vol. 57, No. 4, 1005-1018
https://doi.org/10.4134/JKMS.j190552
Published online July 1, 2020
Mehdi Rafie-Rad, Azadeh Shirafkan
University of Mazandaran; University of Mazandaran
Abstract : A characterization of the $C$-projective vector fields on a Randers space is presented in terms of ${\bf\Xi}$-curvature. It is proved that the ${\bf\Xi}$-curvature is invariant for $C$-projective vector fields. 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.
Keywords : Randers metric, projective vector field, ${\bf S}$-curvature, ${\bf \Xi}$-curvature
MSC numbers : Primary 53C60; Secondary 58B40
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