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Framed curves in the Euclidean space

  • Shun’ichi Honda EMAIL logo and Masatomo Takahashi
From the journal Advances in Geometry

Abstract

A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

MSC 2010: 58K05; 53A04; 53D35

Communicated by: K. Ono


References

[1] V. I. Arnol’d, Singularities of caustics and wave fronts, volume 62 of Mathematics and its Applications (Soviet Series). Kluwer 1990. MR1151185 (93b:58019) Zbl 0734.5300110.1007/978-94-011-3330-2Search in Google Scholar

[2] V. I. Arnol’d, S. M. Guseĭĭn-Zade, A. N. Varchenko, Singularities of differentiable maps. Vol. I. Birkhäuser 1985. MR777682(86f:58018) Zbl 0554.5800110.1007/978-1-4612-5154-5Search in Google Scholar

[3] R. L. Bishop, There is more than one way to frame a curve. Amer. Math. Monthly82 (1975), 246–251. MR0370377 (51#6604) Zbl 0298.5300110.1080/00029890.1975.11993807Search in Google Scholar

[4] M. Bôcher, Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence. Trans. Amer. Math. Soc. 2 (1901), 139–149. MR1500560 JFM 32.0313.0210.1090/S0002-9947-1901-1500560-5Search in Google Scholar

[5] J. W. Bruce, P. J. Giblin, Curves and singularities. Cambridge Univ. Press 1992. MR 1206472(93k:58020) Zbl 0770.5300210.1017/CBO9781139172615Search in Google Scholar

[6] T. Fukunaga, M. Takahashi, Existence and uniqueness for Legendre curves. J. Geom. 104 (2013), 297–307. MR 3089782 Zbl 1276.5300210.1007/s00022-013-0162-6Search in Google Scholar

[7] C. G. Gibson, Elementary geometry of differentiable curves. Cambridge Univ.Press 2001. MR 1855907 (2002i:53005) Zbl 1031.5300210.1017/CBO9781139173377Search in Google Scholar

[8] A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica. Chapman & Hall/CRC, Boca Raton, FL 2006. MR 2253203 (2007d:53001) Zbl 1123.53001Search in Google Scholar

[9] G. Peano, Sur le déterminant Wronskien. Mathesis9(1889), 75–76, 110–112. Zbl 21.0153.01Search in Google Scholar

[10] K. Wolsson, A condition equivalent to linear dependence for functions with vanishing Wronskian. Linear Algebra Appl. 116 (1989), 1–8. MR 989712(90i:34012) Zbl 0671.1500510.1016/0024-3795(89)90393-5Search in Google Scholar

Received: 2014-3-26
Revised: 2014-10-17
Published Online: 2016-6-28
Published in Print: 2016-7-1

© 2016 by Walter de Gruyter Berlin/Boston

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