Directed immersions of closed manifolds

TL;DR

This paper constructs smooth, closed, orientable hypersurfaces in Euclidean space with Gauss maps avoiding specified finite subsets of the sphere, answering a question posed by Gromov.

Contribution

It provides an explicit construction method for immersed hypersurfaces with prescribed Gauss map properties, advancing understanding of geometric immersions.

Findings

Constructed hypersurfaces with Gauss maps missing given sets

Answered Gromov's question on immersions and Gauss maps

Established explicit immersion techniques for closed manifolds

Abstract

Given any finite subset X of the sphere S^n, n>1, 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.