Foundations of singular Finsler geometry
The author offers a singular (non-smooth) generalization of Finsler spaces. Let \(\left( X,d\right) \) be a non-reversible metric space (i.e., the symmetry \(d\left( x,y\right) =d\left( y,x\right) \) need not be true). Then oriented (geodesic) segments are defined via a positive isometry. The negative and positive variations of length, \(V_{-}\left( \gamma \right) \) and \(V_{+}\left( \gamma \right) \), are defined in a standard way with obvious adjustments for metric non-reversibility. A mapping \(\gamma \) is said to be rectifiable if \(d\left( \gamma \left( t\right) ,\gamma \left( t^{\prime }\right) \right) =d\left( \gamma \left( t^{\prime }\right) ,\gamma \left( t\right) \right) \) for all \(t\) and \(t^{\prime }\) and \(\gamma \) is both negatively and positively rectifiable. The space \(\left( X,d\right) \) is said to be geodesic if for every ordered pair of distinct points, there is a directed segment from \(x\) to \(y\). Let \(\gamma ,\gamma ^{\prime }\) be two positively oriented mappings (oriented geodesics) starting at the same point \(x_{0}\in X\). Then the oriented geodesics \(\gamma \) and \(\gamma ^{\prime }\) are said to have the same direction at \(x_{0}\) if both upper limits of \(d\left( \gamma \left( t\right) ,\gamma ^{\prime }\left( t\right) \right) /t\) and \(d\left( \gamma ^{\prime }\left( t\right) ,\gamma \left( t\right) \right) /t\) converge to zero as \(t\rightarrow 0+\). This equivalence relation on the set of all geodesics from \(x_{0}\) defines the set of directions. It is not difficult to introduce the upper isosceles angle between directions -- a straightforward substitute for an upper angle defined via half-angle. The upper isosceles angle defines a non-reversible metric on the set of all directions and determines the space of directions at \(x_{0}\). Finally, the author adopts the notion of a tangent cone (via Hausdorff convergence) for geodesic topologically symmetric (i.e., metric topologies \( \tau _{+}\)and \(\tau _{-}\) coincide) non-reversible metric spaces. The author defines a singular Finsler space as a geodesic topologically symmetric non-reversible metric space such that \(X\) is a topological manifold and such that for every \(x_{0}\in X\), the space of directions is defined independently of the asymmetry, the tangent cone \(K_{x_{0}}X\) is well defined and has a structure of an asymmetric normed linear space with strictly convex norm. He discusses several examples such as Busemann \(G\)-spaces, the Funk metrics and quasi-hyperbolic planes. The author concluded his paper by introducing curves with velocity. He proves that if \(\gamma \) is a curve in a singular Finsler space with existing velocity at every point and such that the norm of the velocity depends continuously on the curve parameter, then \(\gamma \) is positively rectifiable and its positive length is given by the integral of the norm of the velocity.
scientific article; zbMATH DE number 6835951