Gurses' Type (b) Transformations are Neighborhood-Isometries

TL;DR

This paper proves that Gurses' type (b) transformations between Riemannian spaces are neighborhood-isometries, meaning they preserve local geometric structure through specific diffeomorphisms.

Contribution

It establishes that Gurses' type (b) transformations are equivalent to neighborhood-isometries, providing a rigorous geometric interpretation.

Findings

Gurses' type (b) transformations are neighborhood-isometries.

Such transformations preserve local geometric structure.

The result connects Gurses' transformations with generalized diffeomorphisms.

Abstract

Following an idea close to one given by C. G. Torre (private communication), we prove that Riemannian spaces (M,g) and (M,h) that are related by a Gurses type (b) transformation [M. Gurses, Phys. Rev. Lett. 70, 367 (1993)] or, equivalently, by a Torre-Anderson generalized diffeomorphism [C. G. Torre and I. M. Anderson, Phys. Rev. Lett. xx, xxx (1993)] are neighborhood-isometric, i.e., every point x in M has a corresponding diffeomorphism phi of a neighborhood V of x onto a generally different neighborhood W of x such that phi*(h|W) = g|V.