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
References
[1] H. Alencar, W. Santos, G. Silva Neto, Differential geometry of plane curves, volume 96 of Student Mathematical Library. Amer. Math. Soc. 2022. MR4423373 Zbl 1511.5300110.1090/stml/096Search in Google Scholar
[2] V. I. Arnol’d, Topological invariants of plane curves and caustics, volume 5 of University Lecture Series. Amer. Math. Soc. 1994. MR1286249 Zbl 0858.5700110.1090/ulect/005/02Search in Google Scholar
[3] A. A. Borisenko, K. Tenenblat, On the total curvature of curves in a Minkowski space. Israel J. Math. 191 (2012), 755–769. MR3011495 Zbl 1280.5301410.1007/s11856-012-0010-7Search in Google Scholar
[4] K. Borsuk, Sur la courbure totale des courbes fermées. Ann. Soc. Polon. Math. 20 (1947), 251–265 (1948). MR25757 Zbl 0037.23602Search in Google Scholar
[5] F. Brickell, C. C. Hsiung, The total absolute curvature of closed curves in Riemannian manifolds. J. Differential Geometry 9 (1974), 177–193. MR339032 Zbl 0277.5300410.4310/jdg/1214432100Search in Google Scholar
[6] S. S. Chern, Curves and surfaces in Euclidean space. In: Studies in Global Geometry and Analysis, 16–56, Math. Assoc. America 1967. MR212744 Zbl 0683.53002Search in Google Scholar
[7] S.-S. Chern, R. K. Lashof, On the total curvature of immersed manifolds. Amer. J. Math. 79 (1957), 306–318. MR84811 Zbl 0078.1390110.2307/2372684Search in Google Scholar
[8] S.-S. Chern, R. K. Lashof, On the total curvature of immersed manifolds. II. Michigan Math. J. 5 (1958), 5–12. MR97834 Zbl 0095.3580310.1307/mmj/1028998005Search in Google Scholar
[9] K. Enomoto, J.-I. Itoh, R. Sinclair, The total absolute curvature of open curves in E3. Illinois J. Math. 52 (2008), 47–76. MR2507234 Zbl 1202.5300310.1215/ijm/1242414121Search in Google Scholar
[10] I. Fáry, Sur la courbure totale d’une courbe gauche faisant un nœud. Bull. Soc. Math. France 77 (1949), 128–138. MR33118 Zbl 0037.2360410.24033/bsmf.1405Search in Google Scholar
[11] W. Fenchel, Über Krümmung und Windung geschlossener Raumkurven. Math. Ann. 101 (1929), 238–252. MR1512528 Zbl 0257095710.1007/BF01454836Search in Google Scholar
[12] W. Fenchel, On the differential geometry of closed space curves. Bull. Amer. Math. Soc. 57 (1951), 44–54. MR40040 Zbl 0042.4000610.1090/S0002-9904-1951-09440-9Search in Google Scholar
[13] T. Fukunaga, M. Takahashi, On convexity of simple closed frontals. Kodai Math. J. 39 (2016), 389–398. MR3520520 Zbl 1351.5300510.2996/kmj/1467830145Search in Google Scholar
[14] H. Gounai, M. Umehara, Caustics of convex curves. J. Knot Theory Ramifications 23 (2014), 1450050, 28 pages. MR3277959 Zbl 1321.5300610.1142/S0218216514500503Search in Google Scholar
[15] M. Li, G. Wang, ℓ -convex Legendre curves and geometric inequalities. Calc. Var. Partial Differential Equations 62 (2023), Paper No. 135, 24 pages. MR4578453 Zbl 1515.5309810.1007/s00526-023-02480-zSearch in Google Scholar
[16] J. W. Milnor, On the total curvature of knots. Ann. of Math. (2) 52 (1950), 248–257. MR37509 Zbl 0037.3890410.2307/1969467Search in Google Scholar
[17] H. Rutishauser, H. Samelson, Sur le rayon d’une sphère dont la surface contient une courbe fermée. C. R. Acad. Sci. Paris 227 (1948), 755–757. MR27547 Zbl 0036.11206Search in Google Scholar
[18] A. Sama-Ae, A. Phon-on, Total curvature and some characterizations of closed curves in CAT k spaces. Geom. Dedicata 199 (2019), 281–290. MR3928802 Zbl 1414.5100710.1007/s10711-018-0349-ySearch in Google Scholar
[19] S. Shiba, M. Umehara, The behavior of curvature functions at cusps and inflection points. Differential Geom. Appl. 30 (2012), 285–299. MR2922645 Zbl 1250.5300510.1016/j.difgeo.2012.04.001Search in Google Scholar
[20] E. Teufel, On the total absolute curvature of closed curves in spheres. Manuscripta Math. 57 (1986), 101–108. MR866407 Zbl 0589.5300110.1007/BF01172493Search in Google Scholar
[21] E. Teufel, The isoperimetric inequality and the total absolute curvature of closed curves in spheres. Manuscripta Math. 75 (1992), 43–48. MR1156214 Zbl 0754.5300910.1007/BF02567070Search in Google Scholar
[22] Y. Tsukamoto, On the total absolute curvature of closed curves in manifolds of negative curvature. Math. Ann. 210 (1974), 313–319. MR365418 Zbl 0273.5303310.1007/BF01434285Search in Google Scholar
[23] M. Umehara, K. Saji, K. Yamada, Differential geometry of curves and surfaces with singularities, volume 1 of Series in Algebraic and Differential Geometry. World Scientific Publ. Co., Hackensack, NJ 2022. MR4357539 Zbl 1497.5300210.1142/12284Search in Google Scholar
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Articles in the same Issue
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- 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
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Articles in the same Issue
- 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
