Donghe Pei

The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space

We define the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these objects and geometric invariants of curves under the action of the Lorentz group.

SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERBOLIC SPACE

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

The Horospherical Geometry of Submanifolds in Hyperbolic Space

We study some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic

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

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Singularities of Hyperbolic Gauss Maps

-space as an application of the theory of Legendrian singularities.

Umbilicity of space-like submanifolds of Minkowski space

We define the notions of (St1 × Ss2)-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.