Time-like hypersurfaces in de Sitter space and Legendrian singularities

We construct a basic framework for studying the extrinsic differential geometry on time-like hypersurfaces in the de Sitter space from the viewpoint of the theory of Legendrian singularities. As an application, we study the contact of time-like hypersurfaces with flat totally umbilic time-like hypersurfaces in the de Sitter space. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Overdetermined systems of first order partial differential equations with singular solution

We study overdetermined systems of first order partial differential equations with singular solutions. The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.     

Timelike Hypersurfaces in Anti-De Sitter Space from a Contact Viewpoint

We study timelike hypersurfaces in anti-de Sitter space from the viewpoint of the Lagrangian/ Legendrian singularity theory.     

Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

We define the notions of (S _t ¹ × S _s ²)-nullcone Legendrian Gauss maps and S ₊²-nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S ₊²-nullcone Lagrangian Gauss maps, we define the notion of S ₊²-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.     

Spacelike parallels and evolutes in Minkowski pseudo-spheres

We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Preprint] we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.     

MRI of endocervical glandular disorders: three cases of a deep nabothian cyst and three cases of a minimal-deviation adenocarcinoma

The incidence of cervical adenocarcinoma is increasing. Nabothian cysts are a common gynecologic condition; if multiple and/or large cysts are present, it is difficult to differentiate them from a minimal-deviation adenocarcinoma (MDA), which is classified as a special type of cervical adenocarcinoma. We report three cases of deep nabothian cysts and three cases of MDAs. Magnetic resonance imaging (MRI) findings, signs, and symptoms of these cases are described. The absence of a watery discharge and an MR image displaying a round or oval cyst without enhancement after intravenous gadolinium are helpful in the diagnosis of a deep nabothian cyst.     

On caustics of submanifolds and canal hypersurfaces in Euclidean space

We investigate a relationship between the caustics of a submanifold of general dimension and of a canal hypersurface of the submanifold in Euclidean space. As a consequence, these caustics are the same. Moreover, induced Lagrangian immersion germs are Lagrangian equivalent under a suitable condition. In order to show the results, we use the theory of Lagrangian singularity and of Legendrian singularity.