scholarly journals Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

2021 ◽  
pp. 1
Author(s):  
Da Rong Cheng ◽  
Xin Zhou
Keyword(s):  
Geodesic Curvature ◽  
Constant Geodesic Curvature

scholarly journals Submersions and curves of constant geodesic curvature

2019 ◽  
Vol 292 (9) ◽  
pp. 1956-1971
Author(s):  
M. Godoy Molina ◽  
E. Grong ◽  
I. Markina
Keyword(s):  
Geodesic Curvature ◽  
Constant Geodesic Curvature

scholarly journals On Pseudospherical Smarandache Curves in Minkowski 3-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Esra Betul Koc Ozturk ◽  
Ufuk Ozturk ◽  
Kazim Ilarslan ◽  
Emilija Nešović
Keyword(s):  
Geodesic Curvature ◽  
Spacelike Curve ◽  
Constant Geodesic Curvature ◽  
Straight Lines

In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curveα, thenαhas constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves.

scholarly journals Min-max theory for networks of constant geodesic curvature

2020 ◽  
Vol 361 ◽  
pp. 106941 ◽  
Author(s):  
Xin Zhou ◽  
Jonathan J. Zhu
Keyword(s):  
Geodesic Curvature ◽  
Constant Geodesic Curvature

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

The concept of geodesic curvature applied to optical surfaces

2015 ◽  
Vol 35 (4) ◽  
pp. 388-393 ◽  
Author(s):  
Sergio Barbero
Keyword(s):  
Geodesic Curvature ◽  
Optical Surfaces

Geodesic Curvature and Area

1982 ◽  
pp. 194-200
Author(s):  
Lars V. Ahlfors
Keyword(s):  
Geodesic Curvature
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