Lagrangian Curves in a 4-dimensional affine symplectic space
نویسندگان
چکیده
Lagrangian curves in R entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in R and determine Lagrangian geodesics.
منابع مشابه
Symplectic Applicability of Lagrangian Surfaces 3 2
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 e...
متن کاملSymplectic 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 e...
متن کاملSymplectomorphism groups and isotropic skeletons
The symplectomorphism group of a 2–dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of the symplectic 4–manifold (M,ω) into a disjoint union of an isotropic 2–complex L and a disc bundle over a symplectic surface Σ which is Poincare dual to...
متن کاملComplete systems of invariants for rank 1 curves in Lagrange Grassmannians
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically ”the Jacobi equations for extremals”, associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of t...
متن کاملar X iv : m at h . SG / 0 40 44 96 v 2 1 3 Ju l 2 00 4 Symplectomorphism groups and isotropic skeletons
The symplectomorphism group of a 2-dimensional surface S is homotopy equivalent to the orbit of a filling system of curves on S. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of M into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface Σ Poincare dual to a multiple of the form. We show...
متن کامل