# Ricci Tensor in an Einstein Manifold

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 $$n>3$$?