Geodesic Curvature and Area

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

scholarly journals Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces

2021 ◽  
Vol 6 (11) ◽  
pp. 11655-11685
Author(s):  
Tong Wu ◽  
◽  
Yong Wang
Keyword(s):  
Gaussian Curvature ◽  
Smooth Surface ◽  
Geodesic Curvature ◽  
Affine Group ◽  
Smooth Curves ◽  
Characteristic Points ◽  
Curves On Surfaces

<abstract><p>In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $ C^2 $-smooth surface in the generalized affine group and the generalized BCV spaces away from characteristic points and signed geodesic curvature for Euclidean $ C^2 $-smooth curves on surfaces. We get Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces.</p></abstract>

scholarly journals Path planning in a Riemannian manifold using optimal control

2020 ◽  
Vol 17 (12) ◽  
pp. 2050181
Author(s):  
Souma Mazumdar
Keyword(s):  
Optimal Control ◽  
Riemannian Manifold ◽  
Initial Point ◽  
Geodesic Curvature ◽  
Optimal Controls ◽  
Arc Length ◽  
Final Point ◽  
Minimization Principle ◽  
Pontryagin Principle ◽  
Extremal Trajectories

We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time [Formula: see text] and shown to have norm [Formula: see text] which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.

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

On prescribing geodesic curvature on D2D2

2005 ◽  
Vol 60 (3) ◽  
pp. 465-473
Author(s):  
P LIU ◽  
W HUANG
Keyword(s):  
Geodesic Curvature

On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

2015 ◽  
Vol 58 (4) ◽  
pp. 713-722
Author(s):  
Simon Brendle ◽  
Otis Chodosh
Keyword(s):  
Maximum Principle ◽  
Geodesic Curvature ◽  
Sharp Inequality ◽  
Closed Curve ◽  
Closed Curves ◽  
The Maximum Principle ◽  
Curved Manifolds ◽  
Maximum Curvature ◽  
Negatively Curved ◽  
Simply Connected

AbstractMotivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

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