A Natural Representation of Volumes Yields a Remarkable Affine Consequence

TL;DR

This paper explores the deep connection between curvature and linear group actions on surfaces by representing curvature-related quantities through volumes of parallelepipeds, linking classical affine geometry with modern geometric invariants.

Contribution

It introduces a novel volume-based representation of curvature ratios, illuminating the geometric meaning of classical invariants within the framework of affine differential geometry.

Findings

Curvature ratio expressed as a volume function.

Connection between Gaussian curvature and parallelepiped volumes.

Historical link to Tietze's and Klein's geometric invariants.

Abstract

At the beginning of the 20th Century there was a growing interest for the investigation of the action of linear groups on the geometry of surfaces. In that context of ideas, the quest for a connection between curvature and the behaviour of linear groups rose naturally. Pursuing the original thought, we investigate how the geometric meaning of this idea is intimately related to the concept of volume of parallelepiped boxes. We show how the ratio of the Gaussian curvature divided by the fourth power of a certain distance of interest in the geometry of surfaces can be represented as a function of volumes. This geometric description explores the profound meaning of a quantity considered by {\c{T}}i{\c{t}}eica in 1907, in a work that sparked a growing interest in affine differential geometry, as an illustration of Felix Klein's Erlangen Program, in which the quest for geometric invariants was the main point of inquiry.