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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.

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    $\begingroup$ Note that, in the case of dimension $3$, this notion of '$\alpha$-Einstein' is simply the condition that the Ricci tensor have a double eigenvalue (namely, the function $a$). There's not much to say about such metrics in general. They aren't real-analytic in harmonic coordinates in general; for example, if a Riemannian $3$-manifold has full rotational symmetry about some point, then its Ricci tensor will have at least a double eigenvalue everywhere. You could probably write down a flow that has such metrics as fixed points, but it won't have any good properties. $\endgroup$
    – Robert Bryant
    Jul 31, 2017 at 14:07
  • $\begingroup$ You are right. I want any possible answer. Thanks for your comment. $\endgroup$
    – C.F.G
    Aug 1, 2017 at 13:00
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    $\begingroup$ You might consider the "harmonic map - Ricci flow" coupled system (see arxiv.org/pdf/0912.2907.pdf). If the target is $\mathbb{R}$, the fixed points will locally be the same as your "$\alpha$-Einstein" metrics, with the added condition that $\alpha$ is harmonic. Depending on your motivation, adding this assumption might be natural. There is a very important connection with general relativity as well, as discussed in e.g. intlpress.com/site/pub/files/_fulltext/journals/cag/2008/0016/… . $\endgroup$
    – Otis Chodosh
    Aug 14, 2017 at 11:37

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For Q2 I found the following:

In Ricci solitons and real hypersurfaces in a complex space form 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.

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