A note on surfaces in the normal bundle of a curve

Abstract

In 3-dimensional Euclidean space, the geometric fig- ures of a regular curve are completely determined by the curvature function and the torsion function of the curve, and surfaces are the fundamental curved spaces for pioneering study in modern geom- etry as well as in classical differential geometry. In this paper, we de¯ne parametrizations for surface by using parametric functions whose images are in the normal plane of each point on a given curve, and then obtain some results relating the Gaussian curva- ture of the surface with curvature and torsion of the given curve. In particular, we find some conditions for the surface to have either nonpositive Gaussian curvature or nonnegative Gaussian curvature.