On surfaces of finite type in Euclidean $3$-space

Abstract

We prove an extension of T. Takahashi's result on minimal submanifolds in Euclidean spaces and in spheres, and as a corollary obtain support for B. Y. Chen's conjecture which claims that the round spheres are the only compact surfaces of finite type in Euclidean 3-space.

Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

“…Related to this, Dillen, Pas and Verstraelen ( [9]) investigated surfaces in E 3 whose immersions satisfy the condition Gauss map G satisfies the condition…”

HYPERSURFACES IN SPACE FORMS SATISFYING THE CONDITION $L_k x = Ax + b$

“…In [4] and inspired by Garay's extension of Takahashi theorem [18,6,7] and its subsequent generalizations and extensions [8,11,10,12,2,3], the first author jointly with Gürbuz started the study of hypersurfaces in the Euclidean space satisfying the general condition L k x = Ax + b, where A ∈ R (n+1)×(n+1) is a constant matrix and b ∈ R n+1 is a constant vector (we refer the reader to the Introduction of [4] for further details). In particular, the following classification result was given in [ In this paper, and as a natural continuation of the study started in [4], we consider the study of hypersurfaces M n immersed either into the sphere S n+1 ⊂ R n+2 or into the hyperbolic space H n+1 ⊂ R n+2 1 whose position vector x satisfies the condition L k x = Ax + b.…”

Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

“…In particular, if the submanifold is a hypersurface, the Gauss map can be identified with the unit normal vector field to it. Dillen, Pas and Verstraelen ( [7]) studied surfaces of revolution in the three dimensional Euclidean space E 3 such that its Gauss map G satisfies the condition ∆G = AG,where A ∈M at (3, R). Baikoussis and Verstraelen ( [3]) studied the helicoidal surfaces in E 3 .…”