Are different geometries really that different?

TL;DR

This paper introduces the concept of centrogeometry, a unified framework suggesting all geometries are related through deformations, with implications for physics including chronogeometry, mechanics, and cosmology.

Contribution

It proposes centrogeometry as a new unified approach, showing all geometries derive from each other via deformations and exploring their physical significance.

Findings

Centrogeometries are interconnected through general deformations.

Isometries of centrogeometries align with Euclidean diffeomorphisms.

Physical implications in chronogeometry, mechanics, cosmology are discussed.

Abstract

Here is presented a concept of centrogeometry which can be seen as a combination of the concept of point-like observer with an idea of Poincar\'{e}'s that different geometries are principally equivalent. As it is to be shown later, all centrogeometries are obtained from each other by general deformation (i.e. active coordinate transformations). Isometries of centrogeometries are equivalent to those of the Euclidean centrogeometry as described by common diffeomorphisms of the Euclidean spheres. There are discussed physical aspects of centrogeometry in the context of chronogeometry, mechanics and cosmology.