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Show that the set of solutions of the system of equations $$x_1^2+\ldots+x_n^2=1$$ and $$x_1+\ldots+x_n=0$$ is an $(n-2)$-dimensional submanifold of $\mathbb{R}^n$.

I want to take $f(x_1,\ldots,x_n)=(x_1^2+\ldots+x_n^2,x_1+\ldots+x_n)$.

Then the desired set is $f^{-1}(1,0)$. It only remains to show that for every $p\in f^{-1}(1,0)$, $f$ is a submersion at $p$. That is, $Df(p)$ is onto.

But $Df(p)$ is the $2\times n$ matrix $\begin{bmatrix}2x_1 & 2x_2 & \ldots & 2x_n \\1 & 1 & \ldots & 1\end{bmatrix}$. If it were not onto, the matrix would have rank $1$, so $x_1=\ldots=x_n$, impossible from the two equations above.

What are other ways to solve this problem? I would be curious to see. :)

Mika H.
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  • I think that's the best way to do it. – Ian Coley Dec 08 '13 at 21:42
  • You're intersecting an $n-$-sphere with an n-plane thru the origin. Rotate the plane (which does not change "manifoldness", since rotation is a diffeomorphism), so that it goes thru the equator, and you will get an $S^{n-2}$ – user99680 Dec 08 '13 at 21:49

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I would just say that you're intersecting an n-sphere with an n-plane . The intersection is an $S^{n-1}$, which you can see by rotating the plane so that it goes thru the equator of the n-sphere. Edit: Notice that rotation is a diffeomorphism, which preserves manifold properties.

user99680
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The intersection of these surfaces forms a "great sphere" on $\mathbb{S}^{n-1}$ which is "obviously" a submanifold. If you don't like the word ``obviously'' the object is a sphere in a linear subspace of $\mathbb{R}^n.$ Can you show that a submanifold of a submanifold is a submanifold?

Igor Rivin
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