A A question on this paper regarding Ricci flow on space-time



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?
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