Tangent bundle embeddings of manifolds in Euclidean space

Abstract

For a given $n$-dimensional manifold $M^n$ we study the problem of finding the smallest integer $N(M^n$ such that $M^n$ admits a smooth embedding in the Euclidean space ${\mathbb{R}}^N$ without intersecting tangent spaces. We use the Poincaré–Hopf index theorem to prove that $N({\mathbb{S}}^1)=4$, and construct explicit examples to show that $N({\mathbb{S}}^n)\le 3n+3$, where ${\mathbb{S}}^n$ denotes the $n$-sphere. Finally, for any closed manifold $M^n$, we show that $2n+1\le N(M^n)\le 4n+1$.