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I would like to show that $S^n \times S^m$ is a submanifold of $2S^{m+n+1}$. With $2S^{m+n+1}$ we denote the $m+n+1$ sphere with radius $\sqrt{2}$. Here $S^n$ and $S^m$ are the unit spheres. I have trouble with proving this, can somebody help me?

Thanks!

AL123
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  • Please do not post identical (or almost identical) questions. Make a slight edit to your first post, or ask follow-up questions in the comments to answers. – Ted Shifrin Nov 10 '19 at 22:26
  • @TedShifrin The post you link to does not contain a valid answer imo. – J. De Ro Nov 10 '19 at 22:43
  • @EpsilonDelta: That doesn't mean one should just make a new post. Plenty of questions don't get immediate satisfactory attention; but we can't have participants posting the same question repeatedly until they are magically satisfied. I would amplify on the answer that's there, but I figured the person who wrote the answer should have a chance to address issues first. – Ted Shifrin Nov 10 '19 at 22:44
  • Fair enough. But I find it fascinating that both answers don't seem to adress the actual problem. – J. De Ro Nov 10 '19 at 22:45

1 Answers1

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If $x_1^2+\cdots +x_n^2=1$ and $y_1^2+\cdots +y_m^2=1$, then $x_1^2+\cdots +x_n^2+y_1^2+\cdots +y_m^2=2$.

Hagen von Eitzen
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  • Yes, this shows that $S^n \times S^m \subseteq 2S^{n+m+1}$, but this does not make it a submanifold! I guess the OP means here regular submanifold. – J. De Ro Nov 10 '19 at 22:03