General affine differential geometry of surfaces in affine space $A^3$, I: the elliptical case

TL;DR

This paper develops a comprehensive affine differential geometry framework for elliptical surfaces in three-dimensional affine space, introducing minimal moving frames, invariants, and classifying constant curvature surfaces up to affine transformations.

Contribution

It introduces a minimal order moving frame and complete invariants for elliptical surfaces, and classifies constant curvature elliptical surfaces under affine congruence.

Findings

Established a complete system of differential invariants for elliptical surfaces.

Classified elliptical surfaces with constant curvature up to affine congruence.

Provided a foundational framework for affine differential geometry of surfaces.

Abstract

In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to affine congruence.