We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci bounded covering geometry.

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.

No abstract

We define the notions of S 1 t × S 1 s -valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of standard techniques of singularity theory for the Lorentzian lightcone height function.

The main goal of this paper is to study singularities of lightlike surfaces and focal surfaces of spacelike curves in Hyperbolic space times sphere. To do that, we construct a de Sitter height function and a Lightcone height function, and then show the relation between singularities of the lightlike surfaces (respectively, the focal surfaces) and that of the de Sitter height functions (respectively, the Lightcone height functions). In addition, some geometry properties of the spacelike curves are studied from geometric point of view.