Directed immersions of closed manifolds

Abstract

Given any finite subset $X$ of the sphere $S^n$, $n\ge 2$, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space $R^{n+1}$ whose Gauss map misses $X$. In particular, this answers a question of M Gromov.