Spacelike parallels and evolutes in Minkowski pseudo-spheres

Abstract

We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [18] 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.

Global properties of codimension two spacelike submanifolds in Minkowski space

“…It can be shown that γ, γ = −2 (see [15], or [21] for details on this calculation). Since γ lies in LC * , we have that the position vector γ is a parallel lightlike normal field along γ and so is γ .…”

“…In [21] we have defined the notion of the total evolute T E γ of γ : S 1 −→ LC * which is decomposed into T E γ = HE γ ∪ DE γ , where HE γ ⊂ H 2 (−1) and DE γ ⊂ S 2 1 . We have shown that the singularities of the total evolute is corresponding to the flattening points of γ.…”

Pedal foliations and Gauss maps of hypersurfaces in Euclidean space

“…We say that F is a generating family of the graphlike Legendrian unfolding L F (C(F )). We can use all equivalence relations introduced in the previous paper [13,14,15]. Especially, the S.P + -Legendrian equivalence among graphlike Legendrian unfoldings was given in the above contects.…”

Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

“…On the other hand, singularity theory tools are useful in the investigation of geometric properties of submanifolds immersed in different ambient spaces, from both the local and global viewpoint [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The natural connection between geometry and singularities relies on the basic fact that the contacts of a submanifold with the models (invariant under the action of a suitable transformation group) of the ambient space can be described by means of the analysis of the singularities of appropriate families of contact functions, or equivalently, of their associated Lagrangian and/or Legendrian maps.…”