On the pseudo-Riemann's quartics in Finsler's geometry

TL;DR

This paper explores the limitations of traditional Finsler geometry in modeling Riemann's quartic structures, revealing singularities and proposing a weaker geometric framework with potential physical implications.

Contribution

It introduces a weaker Finsler structure applicable to indefinite signatures and analyzes singular hypersurfaces related to Riemann's quartic in physics.

Findings

Finsler's geometry is too restrictive for Riemann's quartic in Euclidean spaces.

Indefinite signature spaces exhibit singular hypersurfaces where Finsler axioms break down.

A weaker Finsler structure is proposed to handle these singularities.

Abstract

An extension of Riemmann's geometry into a direction dependent geometric structure is usually described by Finsler's geometry. Historically, this construction was motivated by the well-known Riemann's quartic length element example. Quite surprisingly, the same quartic expression emerges in solid-state electrodynamics as a basic dispersion relation -- covariant Fresnel equation. Consequently, Riemann's quartic length expression can be interpreted as a mathematical model of a well-established physics phenomena. In this paper, we present various examples of Riemann's quartic that demonstrate that Finsler's geometry is too restrictive even in the case of a positive definite Euclidean signature space. In the case of the spaces endowed with an indefinite (Minkowski) signature, there are much more singular hypersurfaces where the strong axioms of Finsler's geometry are broken down. We propose a weaker definition of Finsler's structure that is required to be satisfied only on open subsets of the tangent bundle. We exhibit the characteristic singular hypersurfaces related to Riemann's quartic and briefly discuss their possible physical interpretation.