ISOMETRIC IMMERSIONS IN 3-DIMENSIONAL EUCLIDEAN SPACE

Abstract

In this paper, we examine the image of geodesic curves of Riemann 2-manifolds under the isometric immersions, in three dimensional Euclidean space. We show that the curvature of these curves is equal to the normal curvature of the manifold in the direction of tangent vector field of the geodesics. Moreover, we prove that if the parameter curves of the manifold are the line of curvature, then the geodesic torsion of geodesics is equal to the torsion of the image curve.