Parallel tangent hyperplanes

Let Sigma(2n) be a smooth strictly convex closed hypersurface in R2n+1 and let M-2n be any oriented smooth connected manifold immersed in R2n+1 : Suppose that f is a continuous function from Sigma(2n) to M-2n. Then there is at least one point p is an element of 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 point corresponding to f(p): If there did 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 our immersion was an embedding, the Euler characteristic of M-2n would have to be equal to +/-2. For any Sigma(2n) and any immersed M-2n we could always get maps f for which the number of points p satisfying the conditions of our theorem exactly equaled two. An example can be given in which both Sigma(2n) and M-2n are the unit sphere about the origin in R2n+1 and there is only one such point p.