On the semidiscrete differential geometry of A-surfaces and K-surfaces

Abstract

In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface’s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.