The functional definition of geodesics

TL;DR

This paper introduces a functional approach to defining geodesics that extends classical concepts to topological manifolds and includes non-linear and generalized geodesics beyond traditional connection-based definitions.

Contribution

It presents a new functional definition of geodesics applicable to topological manifolds, unifying smooth and non-smooth cases, and introduces examples of generalized geodesics not derived from any connection.

Findings

The new definition coincides with classical geodesics in smooth manifolds.

It includes geodesics of non-linear, homogeneous connections.

Provides examples of generalized geodesics not from any connection.

Abstract

In this paper a functional definition of geodesics is introduced which allows to generalize the notion of a geodesic from smooth to topological manifolds. It is shown that in the smooth case the new definition coincides with the classical definition of geodesics of a linear connection. If the smoothness is not required, it is shown by an example that the new definition includes geodesics of non-linear, homogeneous connections. Moreover, an example of generalized geodesics which does not arise from any connection is presented here.