Surface immersions in normed spaces from the affine point of view

V. Balestro, H. Martini and R. Teixeira, Surface immersions in normed spaces from the affine point of view, to appear in Geometriae Dedicata, 2018. (arXiv link) (journal link)

Comments: We’ve wrote a paper developing curvature concepts for surfaces immersed in three-dimensional spaces endowed with a norm (principal curvatures, Gaussian and mean curvatures, normal curvature). It turns out that the idea to do so came from affine differential geometry, and hence we naturally had a lot of questions related to this area within our context. We study affine distance functions, and from this we re-obtain some curvature concepts and also prove some rigidity theorems. We also introduce a Riemannian metric which is closely related to the induced geometry of the surface. Finally, we show in which cases the Birkhoff-Gauss map of a surface is the Blaschke vector field.