Directed immersions of closed manifolds (Q538528)

To every \(C^1\) immersion \(f: M^n \to \mathbb R^{n+1}\) of a closed oriented \(n\)-manifold \(M,\) one can associate the subset \(G_f(M)\) of \(S^n\) known as Gauss image or spherical image of \(M.\) This paper deals with the problem of characterizing the subsets of \(S^n\) which can appear as spherical images of an immersion \(f: M \to \mathbb R^{n+1}.\) It is well known that if the Gauss image is not onto \(S^n,\) \(M\) must be parallelizable; in particular when \(n=2,\) the surface \(M\) is the torus \(T^2.\) Given any finite subset \( X \) of the sphere \(S^n\), \(n \geq 2\), containing no pairs of antipodal points, the author explicitly constructs smoothly immersed closed orientable hypersurfaces \(M\) in Euclidean space \(\mathbb R^{n+1}\) whose Gauss map misses \( X.\) In particular, his result answers a question of M. Gromov.