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Affine classification of \(n\)-curves. (English) Zbl 1161.53015

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).

MSC:

53A15 Affine differential geometry
53A04 Curves in Euclidean and related spaces
53A55 Differential invariants (local theory), geometric objects
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