The third fundamental form metric for hypersurfaces in nonflat space forms

1999 ◽  
Vol 65 (1-2) ◽  
pp. 130-142 ◽  
Author(s):  
H. L. Liu ◽  
U. Simon ◽  
L. Verstraelen ◽  
C. P. Wang
Keyword(s):  
Fundamental Form ◽  
Space Forms ◽  
The Third ◽  
Third Fundamental Form

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

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 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.

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 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 Finite Element Model for Linear Elastic Thick Shells Using Gradient Recovery Method

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Achille Germain Feumo ◽  
Robert Nzengwa ◽  
Joseph Nkongho Anyi
Keyword(s):  
Finite Element ◽  
Fundamental Form ◽  
Element Model ◽  
Shear Stresses ◽  
Gradient Recovery ◽  
Recovery Method ◽  
The Third ◽  
New Family ◽  
Convergence Results ◽  
Third Fundamental Form

This research purposed a new family of finite elements for spherical thick shell based on Nzengwa-Tagne’s model proposed in 1999. The model referred to hereafter as N-T model contains the classical Kirchhoff-Love (K-L) kinematic with additional terms related to the third fundamental form governing strain energy. Transverse shear stresses are computed and C0 finite element is proposed for numerical implementation. However, using straight line triangular elements does not guarantee a correct computation of stress across common edges of adjacent elements because of gradient jumps. The gradient recovery method known as Polynomial Preserving Recovery (PPR) is used for local interpolation and applied on a hemisphere under diametrically opposite charges. A good agreement of convergence results is observed; numerical results are compared to other results obtained with the classical K-L thin shell theory. Moreover, simulation on increasing values of the ratio of the shell shows impact of the N-T model especially on transverse stresses because of the significant energy contribution due to the third fundamental form tensor present in the kinematics of this model. The analysis of the thickness ratio shows difference between the classical K-L theory and N-T model when the ratio is greater than 0.099.

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