Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 098, 27 pages      arXiv:0910.3646      https://doi.org/10.3842/SIGMA.2009.098
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Contact Geometry of Curves

Peter J. Vassiliou
Faculty of Information Sciences and Engineering, University of Canberra, 2601 Australia

Received May 07, 2009, in final form October 16, 2009; Published online October 19, 2009

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 H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.

Key words: moving frames; Goursat normal forms; curves; Riemannian manifolds.

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