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


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?


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