Differential geometry of immersed surfaces in three-dimensional normed spaces

TL;DR

This paper explores the curvature properties of immersed surfaces in three-dimensional Minkowski spaces, defining new curvature concepts via a Birkhoff orthogonality-based Gauss map, and characterizes Minkowski spheres through constant Gaussian curvature.

Contribution

It introduces a novel approach to defining and relating curvature types in Minkowski spaces using a Birkhoff orthogonality-based Gauss map, and characterizes Minkowski spheres via constant Gaussian curvature.

Findings

Defined principal, Gaussian, and mean curvatures using eigenvalues of the differential of the Gauss map.

Established relations between different curvature types in Minkowski spaces.

Proved that compact surfaces with constant Minkowski Gaussian curvature are Minkowski spheres.

Abstract

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere.