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A generalization of Frenet’s frame for non-degenerate quadratic forms with any index. (English) Zbl 1032.53009

Séminaire de théorie spectrale et géométrie. Année 2001-2002. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 20, 101-130 (2002).
A parametrized curve \(c: I\to\mathbb{R}^n\) is called \(r\)-regular if the derivatives \(c^1,\dots, c^{(r)}\) are linearly independent while \(c^{(r+ 1)}\) is a linear combination of \(c^1,\dots, c^{(r)}\). The authors assume \(\mathbb{R}^n\) to be equipped with a nondegenerate quadratic form \(\langle,\rangle\) and that the Gram determinant of \(c^1,\dots,c^{(k)}\) with respect to \(\langle,\rangle\) has constant sign for \(k= 1,2,\dots, r\). They derive canonical Frenet formulas for such curves with respect to \(\langle,\rangle\). The form of the Frenet matrix and of the sequence of (higher) curvatures classify the curves. The authors mention that the construction can be generalized to a (pseudo-)Riemannian manifold by means of the Levi-Civita derivative and parallel transport.
For the entire collection see [Zbl 1008.00009].

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53A04 Curves in Euclidean and related spaces
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