Curves in affine and semi-Euclidean spaces. (English) Zbl 1330.53017

The author introduces the centroaffine invariants for curves in the centroaffine 2-dimensional and 3-dimensional spaces. Like in the Euclidean case such curves are determined by 1 curvature function in the 2-dimensional case, and 2 curvature functions in the 3-dimensional case.
As applications, amongst others, the curves with constant curvature functions, or the curves for which the affine arc length corresponds to the Euclidean arc length are classified.


53A15 Affine differential geometry
53A04 Curves in Euclidean and related spaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI


[1] Chen B.Y.: When does the position vector of a space curve always lie in its rectifying plane. Am. Math. Mon. 110, 147–152 (2003) · Zbl 1035.53003 · doi:10.2307/3647775
[2] Gardner R.B., Wilkens G.R.: The fundamental theorems of curves and hypersurfaces in centroaffine geometry. Bull. Belg. Math. Soc. Simon Stevin 4, 379–401 (1997) · Zbl 0936.53010
[3] Hu N.: Centroaffine space curves with constant curvatures and homogeneous surfaces. J. Geom. 102, 103–114 (2011) · Zbl 1252.53008 · doi:10.1007/s00022-012-0105-7
[4] Laugwitz D.: Differentialgeometrie in Vektorräumen unter besonderer Berücksichtigung der unendlichdimensionalen Räume. Friedr. Vieweg und Sohn, Braunschweig (1965) · Zbl 0139.14904
[5] Li A.M., Simon U., Zhao G.: Global Affine Differential Geometry of Hypersurfaces. W. De Gruyter, Berlin (1993)
[6] Liu, H.: Curves in the lightlike cone. Contrib. Algebr. Geom. 45, 291–303 (2004). Erratum: 54, 469–470 (2013) · Zbl 1063.53001
[7] Liu H.: Ruled surfaces with lightlike ruling in 3-Minkowski space. J. Geom. Phys. 59, 74–78 (2009) · Zbl 1157.53004 · doi:10.1016/j.geomphys.2008.10.003
[8] Liu, H., Meng, Q.: Representation formulas of curves in a 2 and 3 dimensional lightlike cone. Results Math. 59, 437–451 (2011). Erratum: 61, 423–424 (2012) · Zbl 1219.53007
[9] Mayer O., Myller A.: Géométrie centro-affine differentielle des courbes planes. Annales Jassy 18, 234–280 (1933)
[10] Pekşen Ö., Khadjev D.: On invariants of curves in centro-affine geometry. J. Math. Kyoto Univ. 44, 603–613 (2004)
[11] Polyanin A.D., Zaitsev V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. Chapman & Hall/CRC Press, Boca Raton (2003) · Zbl 1015.34001
[12] Salkowski E.: Affne Differentialgeometrie. Walter de Gruyter, Berlin (1934)
[13] Saǧiroǧlu Y., Pekşen Ö.: The equivalence of centro-equiaffine curves. Turk. J. Math. 34(1), 95–104 (2010) · Zbl 1204.53008
[14] Schirokow P.A., Schirokow A.P.: Affne Differentialgeometrie. B. G. Teubner Verlagsgesellschaft, Leipzig (1962) · Zbl 0106.14703
[15] Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University Tokyo. ISBN 3-7983-1529-9 (1991) · Zbl 0780.53002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.