1,720,994 research outputs found
Symplectic Applicability of Lagrangian Surfaces
Full text linkWe 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 considere
Deformation of Surfaces in Lie Sphere Geometry
Full text linkThe theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations, and the corresponding existence problem is discusse
Darboux transform of Dupin Surfaces
No full textWe present a Möbius invariant construction of the Darboux transformation for isothermic surfaces by the method of moving frames and use it to give a complete classification of the Darboux transforms of Dupin surface
Invariant Signatures of Closed Planat Curves
No full textWe prove that any subset of ℝ2 parametrized by a C 1 periodic function and its derivative is the Euclidean invariant signature of a closed planar curve. This solves a problem posed by Calabi et al. (Int. J. Comput. Vis. 26:107-135, 1998). Based on the proof of this result, we then develop some cautionary examples concerning the application of signature curves for object recognition and symmetry detection as proposed by Calabi et a
Tableaux over Lie algebras, integrable systems, and classical surface theory
Full text linkStarting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G(o)-system of Terng and the curved flat system of Ferus-Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss-Codazzi equations are associated with tableaux over sl(4, R) are discussed
A class of overdetermined systems defined by tableaux: Involutiveness and the Cauchy problem
No full textThis article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux
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