66 research outputs found
The Shi arrangement of the type
In this paper, we give a basis for the derivation module of the cone over the
Shi arrangement of the type explicitly
Cusps of Bishop Spherical Indicatrixes and Their Visualizations
The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspond to the point where Bishop curvatures and type-2 Bishop curvatures vanished and their derivatives are not equal to zero. As applications and illustration of the main results, two examples are given
Envelopes of legendre curves in the unit spherical bundle over the unit sphere
In this paper, we introduce a one-parameter family of Legendre curves in the unit spherical bundle over the unit sphere and the curvature. We give the existence and uniqueness theorems for one-parameter families of spherical Legendre curves by using the curvatures. Then we define an envelope for the one-parameter family of Legendre curves in the unit spherical bundle. We also consider the parallel curves and evolutes of one-parameter families of Legendre curves in the unit spherical bundle and their envelopes. Moreover, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane
Spacelike surfaces in Anti de Sitter four-space from a contact viewpoint
We define the notions of -nullcone Legendrian Gauss
maps and -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 -nullcone Lagrangian Gauss maps, we define the notion
of -nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type
theorem as a global property.
We also introduce the notion of horospherical Gauss maps whch has different geometric properties of the above Gauss maps.
As a consequence, we can say that Anti de Sitter space has much more rich geometric properties than the other space forms such as
Euclidean space, Hyperbolic space, Lorentz-Minkowski space and de Sitter space
The horospherical geomoetry of submanifolds in hyperbolic space
We study some geometrical properties associated to the contact of submanifolds with hyperhorospheres in
hyperbolic -space
as an application of the theory of Legendrian singularities
Special non-lightlike ruled surfaces in Minkowski 3-space
In this paper, we give the existence and uniqueness theorems for non-lightlike framed surfaces and define a special non-lightlike ruled surface in Minkowski 3-space. It may have singularities. We give the conditions for identifying cross-caps and surfaces as developable and maximal. Besides, we demonstrate that if the spacelike ruled surface is developable, then the z-parameter curve is an asymptotic curve if and only if the ruled surface is maximal
Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space
In the present paper, we defined the circular evolutes and involutes for a given spacelike framed curve with respect to Bishop directions in Minkowski 3-space. Then, we studied the essential duality relations among parallel curves, normal surfaces, and circular evolutes and involutes. Furthermore, we also studied the duality relations of their singularities. Based on these studies, we found that it is crucially important to consider the duality relations among different geometric objects for the research of submanifolds with singularities
Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3-Space
Singularities of the focal surfaces and the binormal indicatrix associated with a null Cartan curve will be investigated in Minkowski 3-space. The relationships will be revealed between singularities of the above two subjects and differential geometric invariants of null Cartan curves; these invariants are deeply related to the order of contact of null Cartan curves with tangential planar bundle of lightcone. Finally, we give an example to illustrate our findings
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