Ribaucour surfaces of harmonic type

2021 ◽  
Author(s):  
Armando M. V. Corro ◽  
Carlos M. C. Riveros ◽  
Karoline V. Fernandes
Keyword(s):  
Holomorphic Functions ◽  
Parameter Family ◽  
Isolated Singularities ◽  
Type Representation ◽  
Lines Of Curvature ◽  
Weierstrass Type Representation

We introduce the class of Ribaucour surfaces of harmonic type (in short HR-surfaces) that generalizes the Ribaucour surfaces related to a problem posed by Élie Cartan. We obtain a Weierstrass-type representation for these surfaces which depends on three holomorphic functions. As application, we classify the HR-surfaces of rotation, present examples of complete HR-surfaces of rotation with at most two isolated singularities and an example of a complete HR-surface of rotation with one catenoid type end and one planar end. Also, we present a 5-parameter family of cyclic HR-surfaces foliated by circles in non-parallel planes. Moreover, we classify the isothermic HR-surfaces with planar lines of curvature.

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.

Classes of generalized Weingarten surfaces in the Euclidean 3-space

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Diogo G. Dias ◽  
Armando M. V. Corro
Keyword(s):  
Fixed Point ◽  
Distance Function ◽  
Special Class ◽  
Support Function ◽  
The Other ◽  
Isolated Singularity ◽  
Type Representation ◽  
Weingarten Surfaces ◽  
Parallel Surfaces ◽  
Weierstrass Type Representation

AbstractWe present surfaces with prescribed normal Gaussmap. These surfaces are obtained as the envelope of a sphere congruencewhere the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coefficients are functions that depend on the support function and the distance function from a fixed point (in short, DSGW-surfaces). The linear Weingarten surfaces, Appell’s surfaces and Tzitzeica’s surfaces are all DSGW-surfaces. From them we obtain new classes of DSGWsurfaces applying inversions, dilatations and parallel surfaces. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSG Wsurfaces). As applications we classify the EDSGW-surfaces of rotation and present a 2-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes.

A study on timelike circular surfaces in Minkowski 3-space

2020 ◽  
Vol 17 (06) ◽  
pp. 2050074
Author(s):  
Rashad A. Abdel-Baky ◽  
Nadia Alluhaibi ◽  
Akram Ali ◽  
Fatemah Mofarreh
Keyword(s):  
Geodesic Curvature ◽  
Parameter Family ◽  
Lines Of Curvature ◽  
Constant Radius ◽  
Spherical Indicatrix ◽  
New Type ◽  
Circular Surface ◽  
Fixed Radius

This paper studies a smooth one-parameter family of standard Lorentzian circles with fixed radius. Such a surface is called a timelike circular surface with constant radius. We call each circle a generating circle. A new type of timelike circular surfaces was identified and coined as the timelike tangent circular surface. The new timelike tangent circular surface has the property of all generating circles being lines of curvature and its Gaussian and mean curvatures being independent of the geodesic curvature of the spherical indicatrix.

scholarly journals Pseudospherical surfaces of low differentiability

2016 ◽  
Vol 16 (1) ◽  
pp. 1-20
Author(s):  
Josef F. Dorfmeister ◽  
Ivan Sterling
Keyword(s):  
Gauss Curvature ◽  
Type Representation ◽  
Pseudospherical Surfaces ◽  
Weierstrass Type Representation

AbstractWe continue our investigations into Toda’s algorithm [14; 3], which gives a Weierstrass-type representation of Gauss curvature K = −1 surfaces in R

Sign in / Sign up

Export Citation Format

Share Document