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 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 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).

scholarly journals CURVES ON RULED SURFACES UNDER INFINITESIMAL BENDING

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Marija Najdanović ◽  
Miroslav Maksimović ◽  
Ljubica Velimirović
Keyword(s):  
Euclidean Space ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
Hyperbolic Paraboloid ◽  
3 Dimensional ◽  
Infinitesimal Bending

Infinitesimal bending of curves lying with a given precision on ruled surfaces in 3-dimensional Euclidean space is studied. In particular, the bending of curves on the cylinder, the hyperbolic paraboloid and the helicoid are considered and appropriate bending fields are found. Some examples are graphically presented.

Complete 3-dimensional λ-translators in the Euclidean space ℝ4

2021 ◽  
pp. 1-54
Author(s):  
Zhi Li ◽  
Guoxin Wei ◽  
Gangyi Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
3 Dimensional ◽  
The Second Fundamental Form

In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].

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 On Various Definitions of Capacity and Related Notions

1967 ◽  
Vol 30 ◽  
pp. 121-127 ◽  
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Euclidean Space ◽  
Positive Charge ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Electric Capacity ◽  
De La Vallée Poussin

The electric capacity of a conductor in the 3-dimensional euclidean space is defined as the ratio of a positive charge given to the conductor and the potential on its surface. The notion of capacity was defined mathematically first by N. Wiener [7] and developed by C. de la Vallée Poussin, O. Frostman and others. For the history we refer to Frostman’s thesis [2]. Recently studies were made on different definitions of capacity and related notions. We refer to M. Ohtsuka [4] and G. Choquet [1], for instance. In the present paper we shall investigate further some relations among various kinds of capacity and related notions. A part of the results was announced in a lecture of the author in 1962.

Sign in / Sign up

Export Citation Format

Share Document