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The Ricci flow deforms a Riemannian metric. I was wondering if there was something very similar which deforms a pseudo-Riemannian metric or if not, is there reason why such a geometric flow cannot exist?

Also, why in particular does the Ricci flow only work with a Riemannian metric? What would fail in particular for the pseudo-Riemannian case?

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    $\begingroup$ The flow is parabolic only if the metric is positive definite, because the linearized Ricci operator is essentially the Laplacian. If the metric is not positive definite, then the corresponding Laplacian is not elliptic. Such operators, as well as the corresponding flow equation are poorly understood. $\endgroup$
    – Deane Yang
    Commented Jul 4, 2019 at 3:40
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    $\begingroup$ But literally, yes, there is a Ricci flow. The same differential equation makes sense for pseudo-Riemannian metrics. Deane Yang is answering the more subtle questions what is known about it and is it of any use? I haven't worked out any examples in the pseudo-Riemannian metric case but my first guess is the manifold would tend to collapse in the lightlike or spacelike direction. $\endgroup$
    – Ryan Budney
    Commented Jul 4, 2019 at 15:11
  • $\begingroup$ @Ryan Budney: That's interesting, do you think there might be any applications to general relativity in using the Ricci flow for a Lorentzian metric? Something which can be modelled by a spacetime which collapses in the lightlike direction etc. $\endgroup$
    – Hollis Williams
    Commented May 6, 2020 at 19:04

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