The total absolute curvature of closed curves with singularities
-
Atsufumi Honda
,
Chisa Tanaka
Abstract
In this paper, we give a generalization of Fenchel’s theorem for closed curves as frontals in Euclidean space ℝn. We prove that, for a non-co-orientable closed frontal in ℝn, its total absolute curvature is greater than or equal to π. It is equal to π if and only if the curve is a planar locally L-convex closed frontal whose rotation index is 1/2 or –1/2. Furthermore, if the equality holds and if every singular point is a cusp, then the number N of cusps is an odd integer greater than or equal to 3, and N = 3 holds if and only if the curve is simple.
Funding statement: The first author is supported by JSPS KAKENHI Grant Numbers 19K14526, 20H01801 from the Japan Society for the Promotion of Science.
Acknowledgements
The authors would like to thank Masaaki Umehara, Kazuyuki Enomoto and the reviewer for valuable comments.
Communicated by: T. Leistner
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
Artikel in diesem Heft
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
