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 $N(\mathbb{S}^n)\leq 3n+3$, where $\mathbb{S}^n$ denotes the $n$-sphere. Finally, for any closed manifold $M^n$, we show that $2n+1\leq N(M^n)\leq 4n+1$.