Parallel tangent hyperplanes

Let Σ 2 n \Sigma ^{2n} be a smooth strictly convex closed hypersurface in R 2 n + 1 R^{2n+1} and let M 2 n M^{2n} be any oriented smooth connected manifold immersed in R 2 n + 1 . R^{2n+1}. Suppose that f f is a continuous function from Σ 2 n \Sigma ^{2n} to M 2 n . M^{2n}. Then there is at least one point p ∈ Σ 2 n p \in \Sigma ^{2n} such that the hyperplane tangent to Σ 2 n \Sigma ^{2n} at p p is parallel to the hyperplane tangent to the immersed manifold M 2 n M^{2n} at the point corresponding to f ( p ) . f(p). If there did not exist at least two such points, M 2 n M^{2n} would have to be compact and the Hurewicz homomorphism of π 2 n ( M 2 n ) \pi _{2n}(M^{2n}) into H 2 n ( M 2 n ) \mbox {H}_{2n}(M^{2n}) would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of M 2 n M^{2n} would have to be equal to ± 2. \pm 2. For any Σ 2 n \Sigma ^{2n} and any immersed M 2 n M^{2n} we could always get maps f f for which the number of points p p satisfying the conditions of our theorem exactly equaled two. An example can be given in which both Σ 2 n \Sigma ^{2n} and M 2 n M^{2n} are the unit sphere about the origin in R 2 n + 1 R^{2n+1} and there is only one such point p p .