Cross-Ratio Invariants for Surfaces in 4-Space

Abstract

We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas's work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of orthogonal projections of the surface to 3-space. Cross-ratio invariants at P 3 (c)-points are used to recover two moduli in the 4-jet of a projective parametrization of the surface and identify the stable configurations of the asymptotic curves of the surface.

Geometry of surfaces in $$\mathbb R^5$$ through projections and normal sections

“…When projecting a regular surface in R 4 along the unique asymptotic direction u at a parabolic point, the projection has a P 3 (c)singularity at isolated points on the parabolic set, i.e. it is A -equivalent to (x, y) → (x, x 2 y + cy 4 , xy + y 3 ) with c ∈ R \ {0, 1 2 , 1, 3 2 } (see [9]). In particular, the unique asymptotic direction is tangent to the parabolic set of S η at a P 3 (c)-point.…”