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I must prove that an hypersurface $M$ on $\mathbb{R}^{n+1}$ that is Einstein and compact can be only the $n-$dimensional sphere when $n>2$

The Einstein condition we permits to say that scalar curvature of $M$ is costant because $n>2$. The fact that it is an hypersfurface of $\mathbb{R}^{n+1}$ can be use to consider the gauss equation and the Codazzi-Mainardi equation. The fist equation we say that

$H^2-|h|^2=cost=R^M$

Where $H$ is the mean curvature and $h$ is the second fundamental form of $M$.

In the case in which $n=3$ we have that

$cost=H^2-|h|^2=2\lambda\mu=2\det(h)$

So the Gaussian curvature of $M$ is costant but $M$ is compact so it is the 2-sphere on $\mathbb{R}^3$.

How can conclude in the case in which $n>3$?

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