A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such
$$\rho=ag+b\alpha\otimes\alpha$$
where $a,b$ are smooth functions on $M$ and $\rho$ is ricci tensor of $g$. It is easy to see that if $b=0$ then $(M,g)$ reduce to Einstein manifold.


**Q1:** Is there a similar version of the following known theorem such that we deduce $(M,g)$ is a $\alpha-$Einstein:



> 
> **Theorem:** Suppose $(M,g)$ is any Riemannian $n$-manifold with constant
>  sectional curvature $C$. The curvature endomorphism, Ricci tensor, and scalar curvature of $g$ are given by the formulas
>  $$R(X,Y)Z=C(g(Y,Z)X-g(X,Z)Y)$$
>  $$\rho=(n-1)Cg$$
>  $$r=n(n-1)C.$$
> 
> 
> 


**Q2:** As you know Einstein metrics are fixed points of *normalized ricci flow* equation. Is there a PDE such as normalized ricci flow for $\alpha-$Einstein metrics?


Thanks.



Question on $\alpha-$Einstein manifolds
For **Q2** I found the following:


In [Ricci solitons and real hypersurfaces in a complex space form](https://www.jstage.jst.go.jp/article/tmj/61/2/61_205/_article) Cho and Kimura studied on Ricci solitons of real hypersurfaces in a non-flat complex space form and they defined $\alpha-$Ricci soliton $(g,V,\lambda,\mu,\alpha)$, which satisfies the equation
$$L\_V g + 2S + \lambda g + 2\mu\alpha\otimes\alpha = 0,$$
where $\lambda$ and $\mu$ are real constants. In general, we can use this as generelization of $\alpha−$Einstein metric. 


