HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

2013 ◽  
Vol 24 (14) ◽  
pp. 1350107 ◽  
Author(s):  
DAE WON YOON ◽  
DONG-SOO KIM ◽  
YOUNG HO KIM ◽  
JAE WON LEE
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Smooth Functions ◽  
Constant Gaussian Curvature ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Prescribed Gaussian Curvature

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

Examples of surfaces of constant mean curvature

2019 ◽  
pp. 148-154
Author(s):  
M. A. Cheshkova
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Elliptic Integrals ◽  
Mathematical Software ◽  
Surfaces Of Revolution ◽  
Constant Gaussian Curvature ◽  
Constant Length ◽  
Positive Gaussian Curvature ◽  
Curvature Lines

A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

scholarly journals Helicoidal surfaces with constant mean curvature

1982 ◽  
Vol 34 (3) ◽  
pp. 425-435 ◽  
Author(s):  
Manfredo P. do Carmo ◽  
Marcos Dajczer
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Helicoidal Surfaces

scholarly journals Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
Rafael López ◽  
Esma Demir
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Curvature ◽  
First Case ◽  
Helicoidal Surfaces ◽  
Constant Gauss Curvature

AbstractWe classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.

scholarly journals On the second Gaussian curvature of ruled surfaces in Euclidean $3$-space

2006 ◽  
Vol 37 (3) ◽  
pp. 221-226 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Mean ◽  
Second Gaussian Curvature

In this paper, we mainly investigate non developable ruled surface in a 3-dimensional Euclidean space satisfying the equation $K_{II} = KH$ along each ruling, where $K$ is the Gaussian curvature, $H$ is the mean curvature and $K_{II}$ is the second Gaussian curvature.

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