Total curvature and some characterizations of closed curves in CATk spaces

  • Areeyuth Sama-Ae
  • Aniruth Phon-on
Original Paper


In this paper, we study the characterizations of a closed curve in a CAT(k) space that bounds a geodesic surface which is isometric to the disk bounded by a circle in the model space \(S_k\) with same perimeter.


CAT(kTotal curvature Closed curve 

Mathematics Subject Classification (2000)




The authors would like to thank referees for comments and suggestions, which are very helpful to improve the manuscript. This work is supported by Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani 94000, Thailand.


  1. 1.
    Alekseevskij, D.V., Solodovnikov, A.S., Vinberg, E.B.: Geometry of spaces of constant curvature, Geometry II, space of constant curvature. Encycl. Math. Sci. 29, 6–138 (1993)zbMATHGoogle Scholar
  2. 2.
    Alexander, S.B., Bishop, R.L.: The Fary–Milnor theorem in Hadamard manifolds. Proc. Am. Soc. 126, 3427–3436 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alexander, S.B., Bishop, R.L.: Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above. Differ. Geom. Appl. 9, 67–86 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alexandrov, A.D.: Theory of curves based on the approximation by polygonal lines, Nautch. sess. Leningr. Univer., Tesis dokl. na sekth. matem. nauk. (1946)Google Scholar
  5. 5.
    Alexandrov, A.D., Berestovskii, V.N., Nikolaev, I.G.: Generalized Riemannian spaces. Russ. Math. Survey. 41, 1–54 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alexandrov, A.D., Reshetnyak, Y.G.: General Theory of Irregular Curves. Kluwer Acedemic Publishers, Dordrecht (1989)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ballmann, W.: Lectures on Spaces of Nonpositive Curvature. Birkhäuser, Basel (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    Borsuk, K.: Sur la courbure totale des fermées. Ann. Soc. Polaonnaise. 20, 251–265 (1947)zbMATHGoogle Scholar
  9. 9.
    Brickell, F., Hsiung, C.C.: The total absolute curvature of closed curves in Riemannian manifolds. J. Diff. Geom. 9, 177–193 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, vol. 319. Springer, Berlin (1999)zbMATHGoogle Scholar
  11. 11.
    Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  12. 12.
    Fenchel, W.: Uber krummung und windung geschlossener raumkurven. Math. Ann. 101, 238–252 (1929)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lopez, M.C., Mateos, V.F., Masque, J.M.: Total curvature of curves in Riemannian manifolds. Differ. Geom. Appl. 28, 140–147 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Maneesawarng, C., Lenbury, Y.: Total curvature and length estimate for curves in CAT\((K)\) spaces. Differ. Geom. Appl. 19, 211–222 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Milnor, J.M.: On the total curvature of knots. Ann. Math. 52, 248–257 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reshetnyak, Y.G.: Nonexpanding maps in a space of curvature no greater than \(K\). Sibirskii Mat. Zh. 9, 918–928 (1968)Google Scholar
  17. 17.
    Sama-Ae, A., Kharuwannaphat, W., Maneesawarng, C.: A lower bound for total curvature of a closed curve in a CAT(\(K\)) space. JP J. Geom. Topol. 12, 243–262 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sama-Ae, A., Maneesawarng, C.: Geometry of curves on spheres in CAT(\(K\)) spaces. Southeast Asian Bull. Math. 32, 767–778 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sama-Ae, A.: The lower semi-continuity of total curvature for closed curves in CAT(\(K\)) spaces. JP J. Geom. Topol. 15, 17–33 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Szenthe, J.: On the total curvature of closed curves in Riemannian manifolds. Publ. Math. Debrecen. 15, 99–105 (1968)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Teufel, E.: On the total absolute curvature of closed curves in spheres. Manuscr. Math. 57, 101–108 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Teufel, E.: The isoperimetric inequality and the total absolute curvature of closed curves in spheres. Manuscr. Math. 75, 43–48 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tsukamoto, Y.: On the total curvature of closed curves in manifolds of negative curvature. Math. Ann. 210, 313–319 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer Sciences, Faculty of Science and TechnologyPrince of Songkla UniversityMuangThailand

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