# Tangent bundle embeddings of manifolds in Euclidean space

### Mohammad Ghomi

Georgia Institute of Technology, Atlanta, United States

## 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 $2n+1\leq N(M^n)\leq 4n+1$.

## Cite this article

Mohammad Ghomi, Tangent bundle embeddings of manifolds in Euclidean space. Comment. Math. Helv. 81 (2006), no. 1, pp. 259–270

DOI 10.4171/CMH/51