Helicoidal Surfaces With Pointwise 1-Type Gauss Map

FLAT ROTATIONAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN E4

“…By differentiating the first equation in (27) with respect to e 1 and by using (8), the third equation in (27) and (28), we get…”

“…Otherwise, the pointwise 1-type Gauss map is said to be of the second kind. Surfaces in Euclidean space and in pseudo-Euclidean space with pointwise 1-type Gauss map were recently studied in [7], [8], [10], [11], [12], [13], [14]. Also Dursun and Turgay in [9] gave all general rotational surfaces in E 4 with proper pointwise 1-type Gauss map of the first kind and classified minimal rotational surfaces with proper pointwise 1-type Gauss map of the second kind.…”

CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP

“…Furthermore, several classification theorems on rotational surfaces in E 4 and E 4 2 satisfying (1.2) were given in [14,19,22,23]. Recently, the classifications of helicoidal surfaces in E 3 and E 3 1 with (∆-) pointwise 1-type Gauss map have been obtained in [9,11,12].…”

“…Note that if a = 0, then M becomes a rotational surface with the profile curve α(y) = (φ(y), y, 0) (see Section 3). Therefore, we may assume a = 0, i.e., M is a genuine helicoidal surface ( [9]). By a simple calculation, we obtain…”

Helicoidal Surfaces and Their Gauss Map in Minkowski 3-Space Ii

“…In dealing with submanifolds of a Euclidean or a pseudo-Euclidean space, the Gauss map is a useful tool to examine the character of submanifolds in a Euclidean space. For the last few years, two of the present authors and D. W. Yoon introduced and studied the notion of pointwise 1-type Gauss map in a Euclidean or a pseudo-Euclidean space ( [4], [5], [7], [8]), namely the Gauss map G on a submanifold M of a Euclidean space or a pseudo-Euclidean space is said to be of pointwise 1-type if (…”