  For a surface immersed in a three-dimensional space endowed with a norm
instead of an inner product, one can define analogous concepts of curvature and
metric. With these concepts in mind, various questions immediately appear. The
aim of this paper is to propose and answer some of those questions. In this
framework we prove several characterizations of minimal surfaces, and analogues
of some global theorems (e.g., Hadamard-type theorems) are also derived. A
result on the curvature of surfaces of constant Minkowski width is also given.
Finally, we study the ambient metric induced on the surface, proving an
extension of the classical Bonnet theorem.
