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. … Show more
“…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 .…”
Section: Lightcone Configurations and Carathéodory Type Conjectures Omentioning
confidence: 99%
“…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.…”
Abstract. We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional 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 .…”
Section: Lightcone Configurations and Carathéodory Type Conjectures Omentioning
confidence: 99%
“…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.…”
Abstract. We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski 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.…”
Section: Is a Lagrange Stable If And Only If F Is Anmentioning
The singular point of the Gauss map of a hypersurface in Euclidean space is the parabolic point where the Gauss-Kronecker curvature vanishes. It is well-known that the contact of a hypersurface with the tangent hyperplane at a parabolic point is degenerate. The parabolic point has been investigated in the previous research by applying the theory of Lagrangian or Legendrian singularities. In this paper we give a new interpretation of the singularity of the Gauss map from the view point of the theory of wave front propagations.
“…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.…”
Abstract:In this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.
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