Abstract
In 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space E3 which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we study the Darboux curves from a dynamical viewpoint on special canal surfaces, quadrics and some Darboux cyclides. We also describe the generic behavior of Darboux curves near ridge points (zig-zag and beak-to-beak).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. d’Alembert. Opuscules mathémathiques ou Mémoires sur différens sujets de géométrie, de méchanique, d’optique, d’astronomie. Tome VII, (1761), p. 163.
A. Bartoszek, R. Langevin and P. G. Walczak. Special canal surfaces of S3. Bull. Braz. Math. Soc. New Series, 42(2) (2011), 301–320.
J. W. Bruce, P. J. Giblin and F. Tari. Families of surfaces: focal sets, ridges and umbilics. Math. Proc. Cambridge Philos. Soc., 125(2) (1999), 243–268.
R. Bryant. A duality theorem forWillmore surfaces. Journal of Differential Geometry, 20 (1984), 23–53.
G. Cairns, R. Sharpe and L. Webb. Conformal invariants for curves and surfaces in three dimensional space forms. Rocky Mountain Jour. of Math., 24 (1994), 933–959.
E. Cosserat. Sur les courbes tracées sur une surface et dont la sphère osculatrice est tangente en chaque point à la surface. Note Comptes Rendus Acad. Scien. Paris, 121 (1895), 43–46.
G. Darboux. Des courbes tracées sur une surface, dont la sphère osculatrice est tangente en chaque point à la surface. Note Comptes Rendus Acad. Scien. Paris, tome LXXIII (1872), pp. 732–736.
A. Enneper. Bemerkungen über die Differentialgleichung einer Art von Curven auf Flächen. Göttinger Nachrichten (1891), pp. 577–583.
N. Fenichel. Persistence and Smootheness of Invariant Manifolds of Flows. Indiana University Math. J., 21 (1971-1972), 193–226.
L. A. Florit. Doubly ruled submanifolds in space forms. Bull. Belg. Math. Soc. Simon Stevin, 13 (2006), 689–701.
R. Garcia, R. Langevin and P. Walczak. Darboux curves on surfaces I, to appear in the Journal of the Mathematical Society of Japan.
R. Garcia, R. Langevin and P. Walczak. Foliations making a constant angle with principal directions on ellipsoids. Ann. Polon. Math., 113 (2015), 165–173.
R. Garcia and J. Sotomayor. Differential Equations of ClassicalGeometry, a Qualitative Theory. PublicaçõesMatemáticas, 27◦ Colóquio Brasileiro deMatemática, IMPA, (2009).
A. Gullstrand. Zur Kenntniss der Kreispunkte. Acta Math., 29 (1905), 59–100.
J. Haantjes. Conformal differential geometry. V. Special surfaces. Nederl. Akad. Wetensch. Verslagen, Afd. Natuurkunde, 52 (1943), 322–331.
J. G. Hardy. Darboux lines on surfaces. Amer. Journal ofMathematics, 20 (1898), 283–292.
U. Hertrich-Jeromin. Introduction to Möbius Differential Geometry. London Math. Soc. Lecture Notes, vol. 300 Cambridge University Press (2003).
M. Hirsh, C. Pugh and M. Shub. Invariant Manifolds. Lectures Notes in Math., 583 (1977).
F. Klein. Lectures on Mathematics. Macmillan and company (1894), reprint AMS Chelsea publishing Providence, Rhode Island (2011 and 2000).
R. Langevin and J. O’Hara. Conformal arc-length as 1/2 dimensional length of the set of osculating circles. Comment. Math. Helvetici, 85 (2010), 273–312.
R. Langevin and P. G. Walczak. Conformal geometry of foliations. Geom. Dedicata, 132 (2008), 135–178.
W. Melo and J. Palis. GeometricTheory ofDynamical Systems. New York, Springer Verlag (1982).
G. Monge et Hachette. Application de l’analyse à la géométrie, Première Partie, (1807), pp. 1–57.
E. Musso and L. Nicoldi. Willmore canal surfaces in Euclidean space. Rend. Istit. Mat. Univ. Trieste, 31 (1999), 177–202.
A. Pell. D-lines on Quadrics. Trans. Amer. Math. Soc. vol. 1 (1900), 315–322.
I. R. Porteous. Geometric Differentiation. Cambridge Univ. Press (2001).
M. Ribaucour. Propriétés de courbes tracées sur les surfaces. Note Comptes Rendus Acad. Scien. Paris, Tome LXXX, (1875), pp. 642–645.
R. Roussarie. Modèles locaux de champs et de formes. Astérisque, 30 (1975), pp. 181.
L. A. Santaló. Curvas extremales de la torsion total y curvas-D. Publ. Inst. Mat. Univ. Nac. Litoral. (1941), pp. 131–156.
L. A. Santaló. Curvas D sobre conos. Select Works of L. A. Santaló, Springer Verlag (2009), pp. 317–325.
F. Semin. Darboux lines. Rev. Fac. Sci. Univ. Istanbul (A), 17 (1952), 351–383.
M. Spivak. A Comprehensive Introduction to Differential Geometry, vol. III, Publish of Perish Berkeley (1979).
D. Struik. Lectures on Classical Differential Geometry. Addison Wesley (1950), Reprinted by Dover Collections (1988).
M. A. Tresse. Sur les invariants différentiels d’une surface par rapport aux transformations conformes de l’espace. Note Comptes Rendus Acad. Sci. Paris, 192 (1892), 948–950.
E. Vessiot. Contributions à la géométrie conforme. Cercles et surfaces cerclées. J. Math. Pures Appliqués, 2 (1923), 99–165.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Garcia, R., Langevin, R. & Walczak, P. Darboux curves on surfaces II. Bull Braz Math Soc, New Series 47, 1119–1154 (2016). https://doi.org/10.1007/s00574-016-0207-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-016-0207-1