We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvature and for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.
Abstract. As a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) for n, m ≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.A Grassmannian structure of type (n, m) on a manifold M is, by definition, an isomorphism from the tangent bundle T M of M to the tensor product V ⊗ W of two vector bundles V and W with rank n and m over M respectively. Because of the tensor product structure, we have two null plane bundles with fibres P m−1 ( ) and P n−1 ( ) over M . The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka's normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.
A novel color electrophoretic E-Paper using independently movable colored particles with different threshold fields was proposed for the first time. An a-Si TFT driven two primary colors prototype with red and cyan particles was successfully demonstrated. A full color feasibility was also proved using three primary colored particles (yellow, magenta and cyan), which showed a gamut as wide as that of newsprint.
In the split G2-geometry, we study the correspondence found by E. Cartan between the Cartan distribution and the contact distribution with Monge structure on spaces of five variables. Then the generic classification is given on singularities of tangent surfaces to Cartan curves and to Monge curves via the viewpoint of duality. The geometric singularity theory for simple Lie algebras of rank 2, namely, for A2, C2 = B2 and G2 is established.
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