Differential geometry of immersed surfaces in three-dimensional normed spaces

V. Balestro, H. Martini and R. Teixeira, Differential geometry of immersed surfaces in three-dimensional normed spaces, preprint, 2018. (arXiv link)

Comments: The objective here is to develop an analogous theory to that of immersed surfaces in the Euclidean three-dimensional space for the case where the three-dimensional space is endowed with a general norm (sufficiently smooth and strictly convex, actually). There are two main concerns here: to define curvature and to understand how do the surface inherits the ambient metric. This paper is about the first question. The approach taken is inspired by affine differential geometry: the Birkhoff orthogonality relation given by the norm is used to define a transversal normal field on the immersed surface. This normal field “plays the role” of the Gauss map, and then analogous to principal curvatures are derived. Further on, using plane sections we define analogous to the normal curvature, and relate it to the principal curvatures.