FAMILIES OF SURFACES IN ℝ⁴

Abstract

We study the geometry of surfaces in R 4 associated to contact with hyperplanes. We list all possible transitions that occur on the parabolic and so-called A 3 -set, and analyse the configurations of the asymptotic curves and their bifurcations in generic one-parameter families.

Geometric invariants and principal configurations on spacelike surfaces immersed in ℝ3,1

“…The asymptotic directions of Σ are, in fact, the directions u ∈ T x Σ such that dG x (u) is tangent to some null line of Q (we refer to Sections 1.3 and 1.4 of [19] for the Plücker embedding and the linear geometry of the Klein quadric). This observation is in agreement with the fact that the notion of asymptotic directions at a point on a surface immersed in Euclidean 4-space can be stated in terms of the contact of the surface with certain normal hyperplanes or certain lines, where this contact is of higher order (see [16] or [2], respectively).…”

Surfaces in $\mathbb{R}^4$ and their projections to 3-spaces

“…The B 2 -curve of π v , with v asymptotic, is also the A 3 -set of the height function along the binormal direction associated with v [5]. This curve meets the ∆-set tangentially at isolated points [7] and intersects the S 2 -curve transversally at a C 3 -singularity. At inflection points the ∆-set has a Morse singularity and the configuration of the B 2 and S 2 -curves there is given in [5].…”

Reflexion maps and geometry of surfaces in R^4

“…A standard form for the 2-jet of the surface is (x 2 − y 2 , xy, x, y), or in a less reduced form (f 20 x 2 + f 02 y 2 , g 11 xy, x, y), f 20 f 02 < 0, g 11 = 0 as in [4]. This corresponds to 1-jet F 1 = (f 20 u+f 02 w, g 11 v) (or (u−w, v) in reduced form) .…”