On geometry of the first and second fundamental forms of canal surfaces

Abstract

In this study, we analyze the general canal surfaces in terms of the features flat, II-flat minimality and II-minimality, namely we study under which conditions the first and second Gauss and mean curvature vanishes, i.e. K=0, H=0, K_{II}=0 and H_{II} =0. We give a non-existence result for general canal surfaces in E^3 with vanishing the curvatures K, H, K_{II} and H_{II} except the cylinder and cone.We classify the general canal surfaces for which are degenerate according to their radiuses. Finally we obtain that there are no flat, minimal, II-flat and II-minimal general canal surfaces in the Euclidean 3-space such that the center curve has non-zero curvatures.