scholarly journals Submanifolds of a Euclidean space with homothetic Gauss map

1980 ◽  
Vol 32 (3) ◽  
pp. 531-555 ◽  
Author(s):  
Yosio MUTO
Keyword(s):  
Euclidean Space ◽  
Gauss Map

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 The Principal Curvatures and the Third Fundamental Form of Dini-Type Helicoidal Hypersurface in 4-Space

2020 ◽  
pp. 62-68
Author(s):  
Erhan G¨uler
Keyword(s):  
Euclidean Space ◽  
Characteristic Polynomial ◽  
Fundamental Form ◽  
Shape Operator ◽  
Gauss Map ◽  
Dimensional Euclidean Space ◽  
Operator Matrix ◽  
Principal Curvatures ◽  
The Third ◽  
Third Fundamental Form

We consider the principal curvatures and the third fundamental form of Dini-type helicoidal hypersurface D(u, v, w) in the four dimensional Euclidean space E4. We find the Gauss map e of helicoidal hypersurface in E4. We obtain characteristic polynomial of shape operator matrix S. Then, we compute principal curvatures ki=1;2;3, and the third fundamental form matrix III of D.

scholarly journals On biconservative hypersurfaces in pseudo-Riemannian space forms and their Gauss map

2018 ◽  
Vol 103 (117) ◽  
pp. 223-236 ◽  
Author(s):  
Nurettin Turgay
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Space Form ◽  
Gauss Map ◽  
Space Forms ◽  
Geometrical Properties ◽  
Riemannian Space Forms ◽  
Biconservative Hypersurfaces

We first present a survey about recent results on biconservative hypersurfaces in the Minkowski space E4 1, pseudo-Euclidean space E5 2 and Rieamnnian space-form H4. Then we obtain some geometrical properties of these hypersurface families concerning their mean curvature and Gauss map.

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