Abstract
In the present study, we consider canal surfaces imbedded in an Euclidean space of four dimensions. The curvature properties of these surface are investigated with respect to the variation of the normal vectors and curvature ellipse. We also give some special examples of canal surfaces in E^4. Further, we give necessary and sufficient condition for canal surfaces in E^4 to become superconformal. Finally, the visualization of the projections of canal surfaces in E^3 are presented.
Highlights
In this paper, we study canal surfaces imbedded in 4-dimensional Euclidean space E4
We prove necessary and sufficient condition of canal surfaces to become superconformal in E4
The canal surface M with the parametrization (11) in E4 is superconformal if and only if the equalities h(Xu, Xv we obtain −→B, −→C = 0 and →−B = →−C, which shows that M is superconformal
Summary
Let M be a regular surface in E4 given with the parametrization X(u, v) : (u, v) ∈ D ⊂ E2. The tangent space of M at an arbitrary point p = X(u, v) is spanned by the vectors Xu and Xv. The first fundamental form coefficients of M are computed by. The induced Riemannian connection ∇ on M for any given local vector fields X1, X2 tangent to M , is given by. Let us consider the spaces of the smooth vector fields χ(M ) and χ⊥(M ) which are tangent and normal to M , respectively. The second fundamental map is defined as follows:. This map is well-defined, symmetric and bilinear.
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