Contact Geometry of Curves

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2009
• 1. Verfasser: Vassiliou, P.J.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2009
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149111
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ.

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.