scholarly journals On Pseudo-Umbilical Rotational Surfaces with Pointwise 1-type Gauss Map in $\mathbb{E}^4_2$

2017 ◽  
Author(s):  
Burcu Bektaş ◽  
Elif Özkara Canfes ◽  
Uğur Dursun
Keyword(s):  
Gauss Map ◽  
Rotational Surfaces

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.

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

2016 ◽  
Vol 38 (2) ◽  
pp. 305-316 ◽  
Author(s):  
Ferdag Kahraman Aksoyak ◽  
Yusuf Yayli
Keyword(s):  
Gauss Map ◽  
Rotational Surfaces

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

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.

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