Surfaces with closed Möbius form

Li, Fengjiang; Guo, Zhaoli

doi:10.1016/j.difgeo.2014.12.005

Cited by 7 publications

(3 citation statements)

References 16 publications

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“…Without loss of generality, we assume that γ is parametrized by arc-length. For N 2 , we use the parametrization ϕ = ϕ(s, t) given in (8). If we fix the variable t, the curves ϕ(−, t) are geodesics of N 2 because they are congruent generated by evolving γ under the group {φ t }.…”

Section: Criticality Of the Profile Curvesmentioning

confidence: 99%

“…The skew curvature was also used by Milnor to define a family of differential forms in surfaces ( [18,19]). More recently, surfaces with constant skew curvature have been studied as examples of closed Möbius forms ( [8]). In [26], the authors proved that the class of constant skew curvature surfaces does not contain any Bonnet surface, that is, none of them can be isometrically deformed preserving the mean curvature.…”

Section: Introductionmentioning

confidence: 99%

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Classification of rotational surfaces with constant skew curvature in 3-space forms

López¹,

Pámpano²

2020

Preprint

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In this paper, we classify the rotational surfaces with constant skew curvature in 3-space forms. We also give a variational characterization of the profile curves of these surfaces as critical points of a curvature energy involving the exponential of the curvature of the curve. Finally, we provide a converse process to produce all rotational surfaces with constant skew curvature based on the evolution of a critical curve under the flow of its binormal vector field with prescribed velocity.

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Section: Criticality Of the Profile Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Classification of rotational surfaces with constant skew curvature in 3-space forms

López¹,

Pámpano²

2020

Preprint

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“…Recently, surfaces in Euclidean space with constant skew curvature were investigated [23,29] and the problem of finding revolution surfaces with prescribed skew curvature was solved [10] in the context of a quantum constrained dynamics. In this work, we shall address the problem of finding revolution surfaces with prescribed skew curvature in Lorentz-Minkowski space following similar techniques to those of Ref.…”

Section: Introductionmentioning

confidence: 99%

Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space

da Silva

2018

Preprint

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In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew (S) and mean (H) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that H and S can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed H, the strategy consists in rewriting the equation for H, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for revolution surfaces with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for S, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of H and S.

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Submanifolds with constant Moebius curvature and flat normal bundle

Antas,

Tojeiro

2024

manuscripta math.

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Surfaces with closed Möbius form

Cited by 7 publications

References 16 publications

Classification of rotational surfaces with constant skew curvature in 3-space forms

Classification of rotational surfaces with constant skew curvature in 3-space forms

Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space

Submanifolds with constant Moebius curvature and flat normal bundle

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