JournalscmhVol. 81 , No. 1DOI 10.4171/cmh/51

Tangent bundle embeddings of manifolds in Euclidean space

  • Mohammad Ghomi

    Georgia Institute of Technology, Atlanta, United States
Tangent bundle embeddings of manifolds in Euclidean space cover

Abstract

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.