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”ds2 = (dσ/γ)2 from a non-uniform map of an Euclidean space:γ = #G whenG is a finite group, butds2 is also defined by continuity whenG is an infinite compact group (γ = oo).