6,036 research outputs found
Symplectic Applicability of Lagrangian Surfaces
We develop an approach to affine symplectic invariant geometry of Lagrangian
surfaces by the method of moving frames. The fundamental invariants of elliptic
Lagrangian immersions in affine symplectic four-space are derived together with
their integrability equations. The invariant setup is applied to discuss the
question of symplectic applicability for elliptic Lagrangian immersions.
Explicit examples are considered
Contact Geometry of Curves
Cartan's method of moving frames is briefly recalled in the context of
immersed curves in the homogeneous space of a Lie group . 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
-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 is
described. For the special case in which the isometries of act
transitively, it is shown that the contact geometry provides an explicit
algorithmic construction of the differential invariants for curves in . The
inputs required for the construction consist only of the metric 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
and in a family of constant curvature 3-metrics. It is conjectured that
similar results are possible in other Cartan geometries
- …