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, and several analogues of classical rigidity and global theorems are proven in this extended framework (like, e.g., Hadamard-type theorems and the Bonnet theorem). We investigate whether the normal field of a surface is the (Blaschke) affine normal, proving that this is only true for subsets of the spheres in the Euclidean subcase. Moreover, curvatures of surfa 查看全文>>