Monge-Amp\`ere Systems with Lagrangian Pairs

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables $\geq 3$. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3)

Singularities of improper affine spheres and surfaces of constant Gaussian curvature

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 curvatureand for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.Comment: 20 pages, 1 figure

Singular curves of hyperbolic (4,7)(4, 7)-distributions of type C3C_3

A distribution of rank $4$ on a $7$-dimensional manifold is called a $(4, 7)$-distribution if its local sections generate the whole tangent space by taking Lie brackets once. Singular curves of $(4, 7)$-distributions are studied in this paper. In particular the class of hyperbolic $(4, 7)$-distributions of type $C_3$ is introduced and singular curves are completely described via prolongations for them.Comment: 16 pages 2 figure

Singularities of tangent surfaces in Cartan's split G2-geometry

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

Dn-geometry and singularities of tangent surfaces

The geometric model for Dn-Dynkin diagram is explicitly constructed and associated generic singularities of tangent surfaces are classified up to local diffeomorphisms. We observe, as well as the triality in D4 case, the difference of the classification for D3;D4;D5 and Dn(n ・ 6), and a kind of stability of the classification in Dn for n ! 1. Also we present the classifications of singularities of tangent surfaces for the cases B3;A3 = D3;G2;C2 = B2 and A2 arising from D4 by the processes of foldings and removings

Geometry of D_4 conformal triality and singularities of tangent surfaces

It is well known that the projective duality can be understood in the context of geometry of An-type. In this paper, as D4-geometry, we construct explicitly a flag manifold, its triplefibration and differential systems which have D4-symmetry and conformal triality. Then we give the generic classification for singularities of the tangent surfaces to associated integral curves, which exhibits the triality. The classification is performed in terms of the classical theory on root systems combined with the singularity theory of mappings. The relations of D4-geometry with G2-geometry and B3-geometry are mentioned. The motivation of the tangent surface construction in D4-geometry is provided

D_n-geometry and singularities of tangent surfaces

The geometric model for Dn-Dynkin diagram is explicitly constructed and associated generic singularities of tangent surfaces are classified up to local diffeomorphisms. We observe, as well as the triality in D4 case, the difference of the classification for D3,D4,D5 and Dn(n ≥ 6), and a kind of stability of the classification in Dn for n → ∞. Also we present the classifications of singularities of tangent surfaces for the cases B3,A3 = D3,G2,C2 = B2 and A2 arising from D4 by the processes of foldings and removings

Fibulin-4 and -5, but not Fibulin-2, are Associated with Tropoelastin Deposition in Elastin-Producing Cell Culture

Elastic system fibers consist of microfibrils and tropoelastin. During development, microfibrils act as a template on which tropoelastin is deposited. Fibrillin-1 is the major component of microfibrils. It is not clear whether elastic fiber-associated molecules, such as fibulins, contribute to tropoelastin deposition. Among the fibulin family, fibulin-2, -4 and -5 are capable of binding to tropoelastin and fibrillin-1. In the present study, we used the RNA interference (RNAi) technique to establish individual gene-specific knockdown of fibulin-2, -4 and -5 in elastin-producing cells (human gingival fibroblasts; HGF). We then examined the extracellular deposition of tropoelastin using immunofluorescence. RNAi-mediated down-regulation of fibulin-4 and -5 was responsible for the diminution of tropoelastin deposition. Suppression of fibulin-5 appeared to inhibit the formation of fibrillin-1 microfibrils, while that of fibulin-4 did not. Similar results to those for HGF were obtained with human dermal fibroblasts. These results suggest that fibulin-4 and -5 may be associated in different ways with the extracellular deposition of tropoelastin during elastic fiber formation in elastin-producing cells in culture