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.
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