On curvature of surfaces immersed in normed spaces

TL;DR

This paper extends classical surface curvature concepts to surfaces in three-dimensional normed spaces, providing new characterizations, generalizations of fundamental theorems, and volume estimates in Minkowski geometry.

Contribution

It introduces a framework for defining and analyzing curvature in Minkowski spaces using Birkhoff orthogonality, extending classical differential geometry results.

Findings

Characterization of Minkowski Gaussian curvature via surface areas

Generalizations of Huber, Willmore, Alexandrov, and Bertrand-Diguet-Puiseux theorems

Weyl's tube volume formula and curvature-based volume estimates

Abstract

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand-Diguet-Puiseux are derived. A generalization of Weyl's formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtained, and in addition we discuss also two-dimensional subcases of the results in more detail.