Abstract
We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’d, V.I.: Singularities of Caustics and Wave Fronts. Mathematics and its Applications, vol. 62. Kluwer, Dordrecht (1990)
Arnol’d V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Basel (1986)
Bruce J.W., Giblin P.J.: Curves and Singularities. A Geometrical Introduction to Singularity Theory. 2nd ed. Cambridge University Press, Cambridge (1992)
Fukunaga, T., Takahashi, M.: Evolutes of Fronts in the Euclidean Plane. Preprint, Hokkaido University Preprint Series, No.1026 (2012)
Gibson, C.G.: Elementary Geometry of Differentiable Curves. An Undergraduate Introduction. Cambridge University Press, Cambridge (2001)
Gray A., Abbena E., Salamon S.: Modern Differential Geometry of Curves and Surfaces with Mathematica.3rd ed. Studies in Advanced Mathematics. Chapman and Hall, Boca Raton (2006)
Ishikawa G.: Classifying singular Legendre curves by contactomorphisms. J. Geom. Phys. 52, 113–126 (2004)
Saji K., Umehara M., Yamada K.: The geometry of fronts. Ann. Math. 169, 491–529 (2009)
Sasai T.: Geometry of analytic space curves with singularities and regular singularities of differential equations. Funkcial. Ekvac. 30, 283–303 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Masahiko Suzuki on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Fukunaga, T., Takahashi, M. Existence and uniqueness for Legendre curves. J. Geom. 104, 297–307 (2013). https://doi.org/10.1007/s00022-013-0162-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-013-0162-6