Generalization of Conformal Transformations

TL;DR

This paper extends conformal transformations beyond classical Euclidean spaces to Finsler and hypercomplex spaces using analogical geometries, potentially impacting mathematical physics.

Contribution

It introduces a generalization of conformal transformations to higher-dimensional and non-Euclidean spaces via analogical geometries, broadening their applicability.

Findings

Transformations form a group of transitions between projective geometries.

Examples provided for complex and hypercomplex numbers H_4.

Potential link between generalized transformations and analytical functions.

Abstract

Conformal transformations of a Euclidean (complex) plane have some kind of completeness (sufficiency) for the solution of many mathematical and physical-mathematical problems formulated on this plane. There is no such completeness in the case of Euclidean, pseudo-Euclidean and polynumber spaces of dimension greater than two. In the present paper we show that using the concepts of analogical geometries allows us to generalize conformal transformations not only to the case of Euclidean or pseudo-Euclidean spaces, but also to the case of Finsler spaces, analogous to the spaces of affine connectedness. Examples of such transformations in the case of complex and hypercomplex numbers H_4 are presented. In the general case such transformations form a group of transitions, the elements of which can be viewed as transitions between projective Euclidean geometries of a distinguished class fixed by the choice of metric geometry admitting affine coordinates. The correlation between functions realizing generalized conformal transformations and generalized analytical functions can appear to be productive for the solution of fundamental problems in theoretical and mathematical physics.