Invariant volumes, weakly-Berwald Finsler spaces, and the Landsberg—Berwald problem

I discuss invariance of volume forms on a Finsler space. I show that a Finsler space which admits a vertically-invariant volume form which is invariant by the geodesic spray — a geodesically-invariant vertically-invariant volume form — is weakly Berwald, and deduce that a Landsberg space admits a geodesically-invariant vertically-invariant volume form if and only if the space is a Berwald space. I illustrate the ideas by considering Randers spaces.