Tangent bundle embeddings of manifolds in Euclidean space

  • Mohammad Ghomi

    Georgia Institute of Technology, Atlanta, United States


For a given nn-dimensional manifold MnM^n we study the problem of finding the smallest integer N(MnN(M^n such that MnM^n admits a smooth embedding in the Euclidean space RN\mathbb{R}^N without intersecting tangent spaces. We use the Poincaré--Hopf index theorem to prove that N(S1)=4N(\mathbb{S}^1)=4, and construct explicit examples to show that N(Sn)3n+3N(\mathbb{S}^n)\leq 3n+3, where Sn\mathbb{S}^n denotes the nn-sphere. Finally, for any closed manifold MnM^n, we show that 2n+1N(Mn)4n+12n+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