Given an immersed submanifold , we characterize the vanishing of the normal curvature at a point in terms of the behaviour of the asymptotic directions and the curvature locus at . 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.
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Juan José Nuño Ballesteros, María Carmen Romero Fuster, Contact properties of codimension 2 submanifolds with flat normal bundle. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 799–824