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From Metric Properties to a Rigidity Conjecture on Finsler Manifolds via a Geodesic Detecting Algorithm

Abstract. The aim of this paper is to consider Busemann-type inequalities on Finsler manifolds. We actually formulate a rigidity conjecture: any Finsler manifold which is a Busemann NPC space is Berwaldian. This statement is supported by some theoretical results and numerical examples. The presented examples are obtained by using evolutionary techniques (genetic algorithms) which can be used for detecting geodesics on a large class of not necessarily reversible Finsler manifolds.