Generic affine differential geometry of space curves

Abstract

We study affine invariants of space curves from the viewpoint of singularity theory of smooth functions. With the aid of singularity theory, we define a new equi-affine frame for space curves. We also introduce two surfaces associated with this equi-affine frame and give a generic classification of the singularities of those surfaces.

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

“…In particular, lightlike hypersurfaces, which can be constructed as ruled hypersurfaces over spacelike submanifolds of codimension 2, provide good models for the study of different horizon types ( [7] [35]). Singularity theory tools, as illustrated by several papers appeared along the last two decades, have proven to be useful in the description, from both the local and global viewpoint, of geometrical properties of submanifolds immersed in different ambient spaces ( [2,3,4,5,6,8,13,32,36,37,38,39,41,46,47,48]). The natural connection between Geometry and Singularities relies on the basic fact that the contacts of a submanifold with the models (invariant through 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 ( [1], [40], [42]).…”

Singularities of ruled surfaces in ℝ3

“…We can summarize the results of the above theorem as the following relations by referring to the previous results [3,[10][11][12]15]: {Singularities of generic developable surfaces} {Singularities of generic ruled surfaces}, {Singularities of generic ruled surfaces} = {Singularities of generic C ∞ -mappings}. One of the examples of ruled surfaces with cross-caps is the Plücker conoid which is given by γ(θ) = (0, 0, 2 cos θ sin θ) and δ(θ) = (cos θ, sin θ, 0) (0 θ 2π) (cf.…”

R^2 de Bir n-li Eğri Ailesinin Afin Diferansiyel İnvaryantları

“…Özellikle afin diferansiyel invaryantlar teorisinin tarihi 1920'lere dayanır, bu alandaki en temel eserler Blaschke [2], Su [11] ve Schirokow [10] tarafından oluşturulmuştur. Afin diferansiyel geometri ile ilgili diğer çalışmalar [3][4][5] da bulunabilir.…”

“…[7] de, -boyutlu bir afin uzayda bir eğrinin centro-afin invaryantları incelenmiştir. Ayrıca afin grubun alt gruplarında da eğriler ve invaryantları farklı metotlarla ele alınmıştır [1,4,6,10]. Yine afin diferansiyel geometride, diferansiyel invaryantlar kullanılarak eğrilerin denkliğinin araştırılması başka bir problemdir.…”