Critical points of higher order for the normal map of immersions in \(\mathbb{R}^d\) (Q664672)

Let \(M\) denote a \(k\)-dimensional differentiable manifold immersed in \(\mathbb{R}^{k+n}\). The authors describe the image of the critical points of the normal map \(\nu :NM\to \mathbb{R}^{k+n}\) and its relation to a natural generalization of the ellipse of curvature. After defining the \(r\)-critical points of a smooth map between manifolds, they characterize the 2 and 3-critical points of the normal map, with particular detail in the case of 2-critical points of surfaces in \(\mathbb{R}^{4}\).