Unicorns in Finsler Geometry

By unicorns, I am referring to those mythical single-horned horse-like creatures for which there are only rumoured sightings by a privileged few. A similar situation exists in Finsler differential geometry. There, one has the hierarchy Euclidean ⊂ Minkowskian & Riemannian ⊂ Berwald ⊂ Landsberg among five families of metrics, in which the first two inclusions are known to be proper by virtue of explicit examples. However, no attempt within the last few decades to search for a Landsberg metric which is not Berwald has yielded any published example. Nevertheless, Robert Bryant has announced that in two dimensions, there is an abundance of such metrics (albeit in a generalised sense), depending on two families of functions of two variables; among such, there is a subclass with zero flag curvature, depending on one family of functions of two variables. For this reason and for the sake of simpler prose, we shall from now on refer to such metrics as unicorns in Finsler geometry. The purpose of this article is to point out that, through the use of perturbative analysis, it is possible to obtain explicit metrics which are Landsberg to as high order (in some expansion parameter) as we please, but which are only Berwald to zeroth order. The scheme to be described can be implemented in any dimension, and is independent of the method of exterior differential systems used by Bryant. Metaphorically, these approximate but explicit metrics may be likened to silhouettes of the actual unicorns. Our account will close on the somber note that even such approximations come with a hefty price, because a linearisation stability analysis is required to ascertain whether they are truly silhouettes of unicorns, or of other creatures instead. As we shall explain, the same requisite stability analysis is potentially also a viable method for proving the existence of unicorns in any dimension.