Re: Submanifold

The origin is an isolated point for n > 3 as well: use the AM-GM
inequality to show that all other points of X satisfy
x_1^2 + ... + x_n^2 >= n^(n/(n-2)).

In my problem we can let n > 5 > 3, so according to your reasoning of course it won't be a submanifold.

I can't prove though that the inequality you have stated above is indeed true. Obviously we have that:

x_1^2 + ... + x_n^2 >= n * (x_1 * ... * x_n)^(2/n) = n * (x_1^2 + ... + x_n^2)^(2/n).

But what I should do afterwards?.. (It's probably a silly question with an easy answer - as far as I can remember, inequatilites between four means are studied on the begining of the 1st year at the university).

Chris
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