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Geometriae Dedicata

, 136:111 | Cite as

Total torsion of curves in three-dimensional manifolds

  • Claudia C. Pansonato
  • Sueli I. R. CostaEmail author
Original Paper

Abstract

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in \({\mathbb{R}^3}\) by L. A. Santaló, and some results involving the total torsion of lines of curvature.

Keywords

Total torsion Constant curvature manifolds 

Mathematics Subject Classification (2000)

53A35 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics, CCNEUniversity of Santa Maria – UFSMSanta MariaBrazil
  2. 2.Institute of MathematicsUniversity of Campinas – UnicampCampinasBrazil

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