A NOTE ON SURFACES IN THE NORMAL BUNDLE OF A CURVE

In 3-dimensional Euclidean space, the geometric ¯g- 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 di？？erential 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 ¯nd some conditions for the surface to have either nonpositive Gaussian curvature or nonnegative Gaussian curvature.