Advanced Studies in Pure Mathematics

Evolutes of curves in the Lorentz-Minkowski plane

S. Izumiya, M. C. Romero Fuster, and M. Takahashi

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Abstract

We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz-Minkowski plane. This leads naturally to a well defined evolute associated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz-Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points.

Article information

Source
Singularities in Generic Geometry, S. Izumiya, G. Ishikawa, M. Yamamoto, K. Saji, T. Yamamoto and M. Takahashi, eds. (Tokyo: Mathematical Society of Japan, 2018), 313-330

Dates
Received: 26 March 2016
Revised: 29 June 2016
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538618979

Digital Object Identifier
doi:10.2969/aspm/07810313

Mathematical Reviews number (MathSciNet)
MR3839951

Zentralblatt MATH identifier
07085109

Subjects
Primary: 53A35: Non-Euclidean differential geometry
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
evolute inflection point lightcone frame Lagrangian singularity Legendrian singularity

Citation

Izumiya, S.; Fuster, M. C. Romero; Takahashi, M. Evolutes of curves in the Lorentz-Minkowski plane. Singularities in Generic Geometry, 313--330, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07810313. https://projecteuclid.org/euclid.aspm/1538618979


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