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This article is cited in 2 scientific papers (total in 2 papers)
Contact Geometry of Curves
Peter J. Vassiliou Faculty of Information Sciences and Engineering, University of Canberra, 2601 Australia
Abstract:
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group $G$. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the $G$-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds $(M,g)$ is described. For the special case in which the isometries of $(M,g)$ act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in $M$. The inputs required for the construction consist only of the metric $g$ and a parametrisation of structure group $SO(n)$; the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space $H^3$ and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.
Keywords:
moving frames; Goursat normal forms; curves; Riemannian manifolds.
Received: May 7, 2009; in final form October 16, 2009; Published online October 19, 2009
Citation:
Peter J. Vassiliou, “Contact Geometry of Curves”, SIGMA, 5 (2009), 098, 27 pp.
Linking options:
https://www.mathnet.ru/eng/sigma444 https://www.mathnet.ru/eng/sigma/v5/p98
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Abstract page: | 215 | Full-text PDF : | 50 | References: | 23 |
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