Chemical algebra. V:G-weighted distance extensions and metrics

Abstract

The first formulation of the definition equation of completely-G-invariant distance extensions from the action of a compact group G onto a metric space (E, d) is reminded. A more general equation ($\mathbb{E}$) is then consistently associated to a group G mapped by a numerical functionm and acting on a metric space (E, d) mapped by another continuous numerical function. A solution of ($\mathbb{E}$) is called a “G-weighted distance extension ofd”. A differential form of the equation is derived in order to provide a definition of a “G-weighted metric”ds ² = (dσ/γ)² from a non-uniform map of an Euclidean space:γ = #G whenG is a finite group, butds ² is also defined by continuity whenG is an infinite compact group (γ = oo).