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?
- Note first that R×Sn smoothly embeds in Rn+1 for each n, via (t,p)↦etp.
- Taking the Cartesian product with Rm−1, 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.