Modeling of distance functions on a manifold

We propose a theoretical framework for modeling continuous distance functions satisfying the triangle inequality and the identity property, but not necessarily the conditions of symmetry, uniformity, and nonnegativity. We call these distance functions premetrics and they may refer to any cumulative attribute of curves such as energy expended, travel time, travel cost, or travel distance. Our premetrics are derived from a function F0F0 (called the fundamental function of the premetric) by solving a variational problem. We exemplify our theoretical framework with a simple physical model from which we formulate a fundamental function F0F0, and then an asymmetric and nonpositive definite premetric is obtained.