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

Original · EN

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, KII=0 and HII =0. We give a non-existence result for general canal surfaces in E³ with vanishing the curvatures K, H, KII and HII 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.