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$\newcommand{\Ric}{\operatorname{Ric}}\newcommand{\Iso}{\operatorname{Iso}}$Let $(M,g)$ be a Riemannian manifold with corresponding LC connection and Ricci tensor.

Is there an obvious description of the group $\Ric_g$ of all diffeomorphisms preserving the Ricci tensor and comparison to the isometry group $\Iso_g$. Is $\Iso_g$ a normal subgroup of $\Ric_g$? If it is the case what geometric information does the quotion group$\frac{\Ric_g}{\Iso_g}$ have?

In the particular case $\Iso(g)=\Ric_g$ can one say that the manifold admit at least one Einstein structure?

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    $\begingroup$ If the Ricci tensor vanishes, for example on a CY, then the group $Ric_g$ is the diffeomorphism group, and $Iso_g$ can't be normal in it, since its conjugates are isometry groups of diffeomorphic metrics. $\endgroup$
    – Ben McKay
    Commented Jun 7 at 17:02
  • $\begingroup$ @BenMcKay what about finit index case:under what condition the isometry groyp is a finit3 index subgroup? $\endgroup$
    – Ali Taghavi
    Commented Jun 7 at 20:20
  • $\begingroup$ @BenMcKay I wonder under what conditions the group $Ric_g$ is a finite dimensional Lie group? and what is a sharp upper bound for its dimension(Inspired by similar result on isometric group). Any way thank youn very much for your attention to the question $\endgroup$
    – Ali Taghavi
    Commented Jun 8 at 13:17
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    $\begingroup$ Certainly the group $\operatorname{Ric}_g$ is a finite dimensional Lie group acting smoothly whenever the Ricci tensor is definite, i.e. is a pseudo-Riemannian metric. (I don't know of a proof other than in my notes on Cartan geometries.) $\endgroup$
    – Ben McKay
    Commented Jun 10 at 20:39
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    $\begingroup$ My notes on Cartan geometries are here: arxiv.org/abs/2302.14457 $\endgroup$
    – Ben McKay
    Commented Jun 11 at 13:33

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