Contact Geometry of curves

Peter Vassiliou

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    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 Poincare 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.
    Original languageEnglish
    Pages (from-to)1-27
    Number of pages27
    JournalSymmetry, Integrability and Geometry: Methods and Applications
    Volume5
    Issue number98
    DOIs
    Publication statusPublished - 2009

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    Contact Geometry
    Curve
    Differential Invariants
    Affine geometry
    Metric
    Equivalence Problem
    Moving Frame
    Group Action
    Homogeneous Space
    Isometry
    Parametrization
    Equivariant
    Half-space
    Poincaré
    Riemannian Manifold
    Curvature
    Invariant

    Cite this

    Vassiliou, Peter. / Contact Geometry of curves. In: Symmetry, Integrability and Geometry: Methods and Applications. 2009 ; Vol. 5, No. 98. pp. 1-27.
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    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 Poincare 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.",
    author = "Peter Vassiliou",
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    Contact Geometry of curves. / Vassiliou, Peter.

    In: Symmetry, Integrability and Geometry: Methods and Applications, Vol. 5, No. 98, 2009, p. 1-27.

    Research output: Contribution to journalArticle

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    AB - 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 Poincare 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.

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