Surfaces with Constant Chebyshev Angle

Abstract

In this paper we consider surfaces with negative Gaussian curvature parametrized by a generalized Chebyshev net with constant Chebyshev angle in the Euclidean 3-space. We characterize these surfaces in terms of a meromorphic function which satisfies a certain differential equation. Moreover, we show that these surfaces have the geometric property that the asymptotic lines have the same sign of geodesic curvature. As an application we obtain for each constant Chebyshev angle a four-parameter family of complete surfaces.