Contact properties of codimension 2 submanifolds with flat normal bundle

Abstract

Given an immersed submanifold $M^n\subset {\mathbb{R}}^{n+2}$, we characterize the vanishing of the normal curvature $R_D$ at a point $p\in M$ in terms of the behaviour of the asymptotic directions and the curvature locus at $p$. We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.