Conformally invariant skew curves for the total skew curvature on surfaces of $$\mathbb {R}^3$$
Year: 2025
Type: article
Abstract: In differential geometry, curvature-based functionals, such as the total Gaussian curvature, the Willmore energy, and the total geodesic torsion, play a central role in both theoretical investigations and practical applications. In this paper, we study geometric properties of the extremal curves for the next functional $$\begin{aligned} \mathfrak {F}:=\int {\sqrt{H^2-K}}ds=\int {\frac{(k_1-k_2)}{2}}ds, \end{aligned}$$ F : = ∫ H 2 - K d s = ∫ ( k 1 - k 2 ) 2 d s , where ds is the arc element on S and $$k_1,k_2$$ k 1 , k 2 are the principal curvatures. First, we establish that a necessary and sufficient condition for a surface to be a Dupin cyclide is that its lines of curvature and the extremal curves of functional $$\mathfrak {F}$$ F intersect at a constant angle. Secondly, we demonstrate that the extremal curves of the functional $$\mathfrak {F}$$ F are invariant under inversion. Finally, we show that the determination of functional extremal curves of $$\mathfrak {F}$$ F for any cone, general cylinder, and surfaces of revolution can be reduced to quadratures.
Authors: F. Narvaez, R. Garcia
Institutions: University of Cauca, Universidade Federal de Goiás
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,090
Award: G2092045252
OpenAlex ID: W4415525452