# Ricci Tensor and Einstein Manifolds

What can we say about an hypersurface Einstein manifolds on $$\mathbb{R}^{n+1}$$ when $$n\geq 3$$ ?

The manifolds is Einstein so Ricci tensor is a multiply of the metric $$g$$ on manifold:

$$Ric=\lambda g$$ where $$\lambda\in C^\infty(M)$$

If $$S$$ is the scalar curvature of $$M$$ then the relationship between $$S$$ and $$\lambda$$ is

$$\lambda=\frac{S}{n}$$

Because the scalar curvature of $$M$$ is the trace of Ricci tensor.

So using the condition that $$n\geq 3$$ and using a contraction of the second Bianchi Identity we have that the scalar curvature of $$M$$ is costant so $$\lambda$$ is costant.

Another condition that we have is that $$M$$ is an hypersurface of $$\mathbb{R}^{n+1}$$ so we can use also the Codazzi-Mainardi Equation ti get some equation between the mean curvature $$H$$ of $$M$$, the metric $$g$$ and the second fondamental formula of $$M$$. What can other we say about $$M$$, it is possible classify it?