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

摘要

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 (E) is then consistently associated to a group G mapped by a numerical function m and acting on a metric space (E, d) mapped by another continuous numerical function. A solution of (E) is called a "G-weighted distance extension of d". 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 when G is a finite group, but ds~2 is also defined by continuity when G is an infinite compact group (γ = ∞).