Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend:

Prove that Πmi=1Sni can be smoothly embedded in a Euclidean space of dimension 1+mi=1ni.

The solution is apparently fairly simple, but I am having trouble getting a start on this problem. Any help?

Answer

  • Note first that R×Sn smoothly embeds in Rn+1 for each n, via (t,p)etp.
  • Taking the Cartesian product with Rm1, we find that Rm×Sn smoothly embeds in Rm×Rn for each m and n.
  • By induction, it follows that R×mi=1Sni smoothly embeds in a Euclidean space of dimension 1+mi=1ni.

The desired statement follows.

Attribution
Source : Link , Question Author : John Miller , Answer Author : Jim Belk

Leave a Comment