Parallel tangent hyperplanes (Q2781305)

Let \(\Sigma^{2n}\) be a strictly convex hypersurface in \(\mathbb R^{2n+1}\) and let \(M^{2n}\) be any oriented smooth connected manifold immersed in \(\mathbb R^{2n+1}\). Suppose that \(f:\Sigma^{2n}\to M^{2n}\) is a continuous map. In this paper, it is proved that there is at least one point \(p\in\Sigma^{2n}\) such that the hyperplane tangent to \(\Sigma^{2n}\) at \(p\) is parallel to the hyperplane tangent to the immersed manifold \(M^{2n}\) at the corresponding point \(f(p)\). If there do not exist at least two such points, \(M^{2n}\) would have to be compact and the Hurewicz homomorphism of \(\pi_{2n}(M^{2n})\) into \(H_{2n}(M^{2n})\) would have to be surjective. If in addition the immersion \(f\) is an embedding, the Euler characteristic of \(M^{2n}\) would have to be equal to \(\pm 2\). Moreover, for any \(\Sigma^{2n}\) and any immersed \(M^{2n}\), the author shows that there is always a map \(f\) for which the number of points \(p\) satisfying the condition above exactly equals two.