On totally real embeddings into \({\mathbb{C}}^ n\) (Q1079239)

An immersion \(f: M^ m\to {\mathbb{C}}^ n\) is totally real if for each \(p\in M\) the tangent space to f(M) at f(p) contains no nontrivial complex linear subspace (so \(m\geq n)\). The main result of this paper states that when \(m=n\geq 3\), any totally real immersion which is regularly homotopic to an embedding is regularly homotopic through a family of totally real immersions to a totally real embedding. Some applications are given, for example it is shown that every compact orientable 3-manifold admits a totally real embedding in \({\mathbb{C}}^ 3\).