# 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 $R_{N}$ without intersecting tangent spaces. We use the Poincaré–Hopf index theorem to prove that $N(S_{1})=4$, and construct explicit examples to show that $N(S_{n})≤3n+3$, where $S_{n}$ denotes the $n$-sphere. Finally, for any closed manifold $M_{n}$, we show that $2n+1≤N(M_{n})≤4n+1$.

## 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