Helicoidal Surfaces of the Third Fundamental Form in Minkowski 3-Space

Abstract

Abstract. We study helicoidal surfaces with the non-degenerate third fundamental form in Minkowski 3-space. In particular, we mainly focus on the study of helicoidal surfaces with light-like axis in Minkowski 3-space. As a result, we classify helicoidal surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form on the surface.

The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

“…Choi et al [22] studied helicoidal surfaces and their Gauss map in Minkowski three-space. See also [23][24][25][26]. Güler, Magid and Yaylı [27] studied the Laplace-Beltrami operator of a helicoidal hypersurface in E 4 .…”

Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

“…(det I) 3/2 {r(r 4 − 1)(r6 + 2r 4 cos(4θ) + r 2 + a 2 )ϕ +r 2 (3r 4 + 2r 2 cos (4θ) − 1)ϕ 3 − 6ar 3 sin(4θ)ϕ 2 +[2r 2 (−5r 6 + r 2 − 3a 2 ) cos (4θ) + r 10 − 8r 6 + 4a 2 r 4 −r 2 − 2a 2 ]ϕ + 2ar 6r 6 − 2r 2 + a 2 sin(4θ)} (det I) 2 ({[3r 11 + 2r 5 (r 4 − 1) cos(4θ) − 4r 7 + r 3 ]ϕ −2ar 5 (r 4 − 1) sin(4θ)}ϕ + [−2r 6 (3r 4 + 1)…”