scholarly journals Rotational Surfaces with Pointwise 1-Type Gauss Map in Pseudo Euclidean Space $\mathbb{E}_{2}^{4}$

2017 ◽  
Author(s):  
Ferdağ Kahraman Aksoyak ◽  
Yusuf Yaylı
Keyword(s):  
Euclidean Space ◽  
Gauss Map ◽  
Rotational Surfaces

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

scholarly journals New Characterizations of the Clifford Torus and the Great Sphere

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1076 ◽  
Author(s):  
Sun Mi Jung ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
Round Sphere ◽  
Clifford Torus ◽  
Great Sphere ◽  
Set Up

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

scholarly journals Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

2018 ◽  
Vol 36 (3) ◽  
pp. 207-217
Author(s):  
Akram Mohammadpouri
Keyword(s):  
Euclidean Space ◽  
Smooth Function ◽  
Mean Curvature ◽  
Gauss Map ◽  
Constant Vector ◽  
Newton Transformation ◽  
Linearized Operator

In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

scholarly journals The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 186
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Characteristic Polynomial ◽  
Fundamental Form ◽  
Shape Operator ◽  
Gauss Map ◽  
Dimensional Euclidean Space ◽  
Operator Matrix ◽  
Geometric Objects

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects.

scholarly journals Timelike rotational surfaces of elliptic, hyperbolic and parabolic types in Minkowski space E41 with pointwise 1-type Gauss map

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 381-392 ◽  
Author(s):  
Burcu Bektaş ◽  
Uğur Dursun
Keyword(s):  
Minkowski Space ◽  
Parabolic Type ◽  
Gauss Map ◽  
Rotational Surface ◽  
Rotational Surfaces

In this work, we focus on a class of timelike rotational surfaces in Minkowski space E41 with 2-dimensional axis. There are three types of rotational surfaces with 2-dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type. We obtain all flat timelike rotational surface of elliptic and hyperbolic types with pointwise 1-type Gauss map of the first and second kind. We also prove that there exists no flat timelike rotational surface of parabolic type in E41 with pointwise 1-type Gauss map.

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