Inflection points and topology of surfaces in 4-space

Abstract

Abstract. We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

Families of Gauss indicatrices on smooth surfaces in pseudo-spheres in the Minkowski 4-space

“…Points where ∆ is singular (generically a Morse singularity A ± 1 ) are labelled inflection points. The generic configurations of the asymptotic curves at inflection points are the same as those for surfaces in R 4 , top figures in Figure 6 and Figure 7 (see [5,7]).…”

Singularities of asymptotic lines on surfaces in R^4

“…Suppose that the origin is a rank-2 simple singular point. Then the singular point is one of the following cases: Next we recall previous results of geometry of surfaces in R 4 (see [5], [13], [14]). Let f : M 2 → R 4 be an embedding, where M is a compact surface.…”

Cross-Ratio Invariants for Surfaces in 4-Space

“…6 the cr-invariants at P 3 (c)-points and obtain, as a consequence of the main theorem above, the stable configurations of the asymptotic curves and of the various curves introduced in this paper. Garcia et al 2000;Little 1969;Mochida et al 1995;Nuño-Ballesteros and Tari 2007;Oset-Sinha and Tari 2015). Let M be a regular surface in the Euclidean space R 4 .…”

“…3 * * The codimension of P 3 (c) is that of its stratum The direction of the kernel of the Hessian of the height functions along a binormal direction is called an asymptotic direction associated to the given binormal direction (Mochida et al 1995). The asymptotic directions are called conjugate directions in Little (1969), and are defined as the directions along θ such that the vector η(θ) is tangent to the curvature ellipse (see also Garcia et al 2000;Mochida et al 1995). If p is not an inflection point, there are 2/1/0 asymptotic directions at p depending on p being a hyperbolic/parabolic/elliptic point.…”

“…At inflection points, the Δ-set has a Morse singularity. For more details on inflection points see Bruce and Nogueira (1998) and Garcia et al (2000).…”