CHEMICAL ALGEBRA .5. G-WEIGHTED DISTANCE EXTENSIONS AND METRICS

The first formulation of the definition equation of completely-G-invar iant distance extensions from the action of a compact group G onto a m etric space (E, d) is reminded. A more general equation (E) is then co nsistently associated to a group G mapped by a numerical function m an d acting on a metric space (E, d) mapped by another continuous numeric al function. A solution of (E) is called a ''G-weighted distance exten sion of d''. A differential form of the equation is derived in order t o provide a definition of a ''G-weighted metric'' ds(2) = (d sigma/gam ma)(2) from a non-uniform map of an Euclidean space: gamma = #G when G is a finite group, but ds(2) is also defined by continuity when Gis a n infinite compact group (gamma = infinity).