Can Ricci Flow be Used in Lorentz Manifolds?

[wLw]

https://arxiv.org/pdf/1812.06239.pdf
In this paper,the authors use ricci flow to construct Lifshitz spaces. But it is known that ricci flow is limited by Riemannian manifold, which has a positive metric. but in this paper the author use ricci flow in a lorentz manifold, whose signature is(-,+,+,+), is not a Riemannian maniflod. and the metric here is $d s^{2}=l^{2}\left[-f_{1}(\lambda, r) d t^{2}+\frac{1}{r^{2}} d r^{2}+f_{3}(\lambda, r) d x_{i} d x^{i}\right], \qquad i=1,2, \ldots D$

My question is: Why the authors can utilize ricci flow in a lorentz space? can you help me?

 

[Jimster41]

Very naive answer here: My read of the wiki on Lorentz Manifold is that it is a special case of a pseudo-Riemannian manifold that has the right non-degenerate properties (and so differentiable algebraic forms) everywhere even though the requirement of positive semi-definiteness is "relaxed".

IOW I think the answer is that the Lorentzian manifold can be shown to be mathematically "close-enough".

I'm trying to answer in the hopes that someone who really knows about Ricci Flow will chime in.

 

[PeterDonis]

it is known that ricci flow is limited by Riemannian manifold

When you say "it is known", do you mean it has been proved as a theorem, or just that you read it someplace like Wikipedia?