Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

TL;DR

This paper investigates the existence of multiple geodesics in manifolds with Randers-Kropina metrics, applying homotopy theory and control system analysis to establish infinitely many solutions under certain conditions.

Contribution

It introduces a novel approach combining Lusternik-Schnirelman theory with homotopy analysis of affine control systems to prove multiple geodesics in Randers-Kropina spaces.

Findings

Proves existence of infinitely many geodesics between two points in non-contractible manifolds.

Connects homotopy theory with control systems for solving geodesic problems.

Establishes results applicable to spacetime models and navigation problems.

Abstract

We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of nonholonomic constraints that the velocity vectors must satisfy, this is achieved thanks to a recent result about the homotopy type of the set of solutions of an affine control system with a controlled drift and related to a corank one, completely nonholonomic distribution of step 2.