scholarly journals Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

2021 ◽  
Author(s):  
Renzo Caddeo ◽  
Irene I. Onnis ◽  
Paola Piu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Classical Result ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Helicoidal Surfaces ◽  
Helicoidal Surface ◽  
Groups Of Isometries ◽  
Two Parameter

AbstractIn this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 226 ◽  
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Dimensional Euclidean Space ◽  
Natural Numbers ◽  
Rotational Surface ◽  
Isometric Deformation ◽  
Helicoidal Surface

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace–Beltrami operator of the rotational surface of value ( 0 , 1 ) .

scholarly journals A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere

2004 ◽  
Vol 76 (4) ◽  
pp. 639-643 ◽  
Author(s):  
Abdênago Barros
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Stereographic Projection ◽  
Inverse Image ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Right

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

scholarly journals TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE

1995 ◽  
Vol 10 (03) ◽  
pp. 337-364 ◽  
Author(s):  
MICHAEL S. ODY ◽  
LEWIS H. RYDER
Keyword(s):  
Mean Curvature ◽  
Sigma Model ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Gauss Map ◽  
Field Vector ◽  
Dimensional Euclidean Space ◽  
Normal Vector ◽  
Constant Mean Curvature Surface ◽  
Spherical Images

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

scholarly journals Integral Formulas for Submanifolds and their Applications

1970 ◽  
Vol 22 (2) ◽  
pp. 376-388 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Integral Formulas ◽  
Closed Hypersurfaces

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

Sign in / Sign up

Export Citation Format

Share Document