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 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 Surfaces of Finite III-type

2021 ◽  
Vol 15 ◽  
pp. 190-194
Author(s):  
Hassan Al-Zoubi
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Principal Curvature ◽  
Gauss Curvature ◽  
Dimensional Euclidean Space ◽  
Surfaces Of Revolution ◽  
3 Dimensional ◽  
The Third ◽  
Third Fundamental Form ◽  
Special Case

In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces with respect to the third fundamental form of the surface. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.

scholarly journals Surfaces of Finite III-type in the Euclidean 3-Space

2021 ◽  
Vol 20 ◽  
pp. 729-735
Author(s):  
Hassan Al-Zoubi ◽  
Farhan Abdel-Fattah ◽  
Mutaz Al-Sabbagh
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gauss Curvature ◽  
Surfaces Of Revolution ◽  
Laplace Operators ◽  
The Third ◽  
Third Fundamental Form

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Then, we introduce the finite Chen type surfaces of revolution with respect to the third fundamental form which Gauss curvature never vanishes.

scholarly journals On O. Bonnet III-isometry of surfaces in three dimensional Euclidean space

1990 ◽  
Vol 49 (1) ◽  
pp. 90-110 ◽  
Author(s):  
Wenmao Yang
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Surfaces Of Revolution ◽  
Principal Curvatures ◽  
Constant Gaussian Curvature ◽  
Geometric Properties ◽  
3 Dimensional ◽  
The Third

AbstractIn this paper we consider O. Bonnet III-isometry (or BIII-isometry) of surfaces in 3-dimensional Euclidean space E3 Suppose a map F: M → M* is a diffeomorphism, and F* (III*) = III, ki(m) = k*i (m*), i = 1, 2, where m ∈ M, m* ∈ M*, m* = F (m), ki and k*i are the principal curvatures of surfaces M and M* at the points m and m*, respectively, III and III* are the third fundmental forms of M and M*, respectively. In this case, we call F an O. Bonnet III-isometry from M to M*. O. Bonnet I-isometries were considered in references [1]-[5].We distinguish three cases about BIII-surfaces, which admits a non-trivial BIII-ismetry. We obtain some geometric properties of BIII-surfaces and BIII-isometries in these three cases; see Theorems 1, 2, 3 (in Section 2). We study some special BIII-surfaces: the minimal BIII-surfaces; BIII-surfaces of revolution; and BIII-surfaces with constant Gaussian curvature; see Theorems 4, 5, 6 (in Section 3).

On the Third Fundamental Form of a Surface

1953 ◽  
Vol 75 (2) ◽  
pp. 298 ◽  
Author(s):  
Philip Hartman ◽  
Aurel Wintner
Keyword(s):  
Fundamental Form ◽  
The Third ◽  
Third Fundamental Form

A class of generalized special Weingarten surfaces

2019 ◽  
Vol 30 (14) ◽  
pp. 1950075
Author(s):  
Armando M. V. Corro ◽  
Diogo G. Dias ◽  
Carlos M. C. Riveros
Keyword(s):  
Fixed Point ◽  
Holomorphic Functions ◽  
Gauss Map ◽  
Type Representation ◽  
The Third ◽  
Weingarten Surfaces ◽  
Lines Of Curvature ◽  
Third Fundamental Form ◽  
Pass Through ◽  
Weierstrass Type Representation

In [Classes of generalized Weingarten surfaces in the Euclidean 3-space, Adv. Geom. 16(1) (2016) 45–55], the authors study a class of generalized special Weingarten surfaces, where coefficients are functions that depend on the support function and the distance function from a fixed point (in short EDSGW-surfaces), this class of surfaces has the geometric property that all the middle spheres pass through a fixed point. In this paper, we present a Weierstrass type representation for EDSGW-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we classify isothermic EDSGW-surfaces with respect to the third fundamental form parametrized by planar lines of curvature. Moreover, we give explicit examples of EDSGW-surfaces and isothermic EDSGW-surfaces.

scholarly journals Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

2017 ◽  
Vol 48 (2) ◽  
pp. 123-134
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Sezai Kiziltug
Keyword(s):  
Fundamental Form ◽  
Three Dimensional ◽  
Algebraic Equations ◽  
The Third ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
Helicoidal Surfaces ◽  
Coordinate Functions ◽  
Third Fundamental Form ◽  
Laplacian Operators

In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space  I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

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