Parallel tangent hyperplanes

Abstract:

Let $\Sigma ^{2n}$ be a smooth strictly convex closed hypersurface in $R^{2n+1}$ and let $M^{2n}$ be any oriented smooth connected manifold immersed in $R^{2n+1}.$ Suppose that $f$ is a continuous function from $\Sigma ^{2n}$ to $M^{2n}.$ Then 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 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 $\mbox {H}_{2n}(M^{2n})$ 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 $\pm 2.$ 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 $R^{2n+1}$ and there is only one such point $p$.