Inflection points and nonsingular embeddings of surfaces in \(\mathbb{R}^5\) (Q1430432)

This paper studies the question of which surfaces admit a nonsingular embedding of order 2 in \({\mathbb{R}}^5\). The approach is to use the family of height functions induced by an embedding to define the notion of asymptotic direction on \(M\). Critical points of asymptotic direction fields can be viewed either as umbilics of height functions or as a singular point of order 2. It is shown that a surface generically embedded in \({\mathbb{R}}^5\) admits at least one and at most five locally defined fields of asymptotic directions. When such a globally defined field exists on a surface \(M\) with nonvanishing Euler number, then \(M\) must have singular points of order 2.