On spherical product surfaces in E3

Abstract

In the present study we consider spherical product surfaces X = alpha circle times beta of two 2D curves in E-3. We prove that if a spherical product surface patch X = alpha circle times beta has vanishing Gaussian curvature K (i.e. a flat surface) then either alpha or beta is a straight line. Further, we prove that if alpha(u) is a straight line and beta(v) is a 2D curve then the spherical product is a non-minimal and flat surface. We also prove that if beta(v) is a straight line passing through origin and alpha(u) is any 2D curve (which is not a line) then the spherical product is both minimal and flat. We also give some examples of spherical product surface patches with potential applications to visual cyberworlds.