Nadjafikhah, Mehdi; Madhipour Sh., Ali Affine classification of \(n\)-curves. (English) Zbl 1161.53015 Balkan J. Geom. Appl. 13, No. 2, 66-73 (2008). The article is a contribution to the equiaffine classification of the so-called \(n\)-curves \(C:[a, b]\to\mathbb{R}^n\) of class \({\mathcal C}^{n+2}\) with \(\text{det}(C',C'',\dots, C^{(n)})> 0\) by a method due to E. Cartan. Hereby a complete system of invariants (not of minimal order of differentiation) is given by the 1-st, \(\dots,(n-1)\)-th special affine curvature \[ \chi_1:=(-1)^{n-1} \text{det}(\ddot C,\dots, C^{[n+1]}),\dots, \chi_{n-1}:= -\text{det}(\dot C,\dots, C^{[n-2]}, C^{[n]}, C^{[n+1]}), \]referred to the special affine arclength parameter \[ \sigma(t):= \int^t_a \text{det}(C', C'',\dots, C^{(n)})^{{2\over n(n-1)}} dt\text{ of }C, \] in the following sense: Two such curves \(C\) and \(\overline C\) are equiaffine equivalent, if and only if the corresponding special affine curvatures of \(C\) and \(\overline C\) coincide. (Note: Many misprints in the formulas of the paper irritate the reader). Reviewer: Kurt Leichtweiß (Stuttgart) MSC: 53A15 Affine differential geometry 53A04 Curves in Euclidean and related spaces 53A55 Differential invariants (local theory), geometric objects Keywords:equiaffine classification of curves in \(\mathbb{R}^n\); complete systems of differential invariants; Maurer-Cartan form of a matrix Lie group PDFBibTeX XMLCite \textit{M. Nadjafikhah} and \textit{A. Madhipour Sh.}, Balkan J. Geom. Appl. 13, No. 2, 66--73 (2008; Zbl 1161.53015) Full Text: arXiv EuDML