An Algebraic Approach to Geodesics via Mobi Spaces

Mobi spaces were introduced by the authors as a possible algebraic axiomatization of spaces in which any two points are connected by a geodesic path. Previous work has focused primarily on the algebraic properties of these structures; here, we return to the original geometric motivation. We present new characterizations of mobi spaces inspired by the important class of smooth manifolds that arise as open subsets of Euclidean (n)-space endowed with a Riemannian metric. We show that such manifolds satisfy the axioms of a mobi space, thereby providing a broad family of natural geometric examples.