462 research outputs found
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 . 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 -distributions of type
A distribution of rank on a -dimensional manifold is called a -distribution if its local sections generate the whole tangent space by
taking Lie brackets once. Singular curves of -distributions are studied
in this paper. In particular the class of hyperbolic -distributions of
type 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
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