Surface immersions in normed spaces from the affine point of view

TL;DR

This paper explores the differential geometry of surfaces in three-dimensional normed spaces using affine geometry, introducing a Riemannian metric, re-deriving curvatures, and characterizing minimal surfaces and normal vector coincidences.

Contribution

It develops a framework for studying immersed surfaces in normed spaces from an affine perspective, including new characterizations of curvatures and conditions for normal vector fields.

Findings

Re-calculation of Minkowski Gaussian and mean curvatures.

Characterization of minimal surfaces via differential equations.

Identification of conditions when affine and Birkhoff normals coincide.

Abstract

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gaussian and mean curvatures. These curvatures are also re-obtained in terms of ambient affine distance functions, and as a consequence we characterize minimal surfaces as the solutions of a certain differential equation. We also investigate in which cases it is possible that the affine normal and the Birkhoff normal vector fields of an immersion coincide, proving that this only happens when the geometry is Euclidean.