# A doubt 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 $$d s^{2}=l^{2}\left[-f_{1}(\lambda, r) d t^{2}+\frac{1}{r^{2}} d r^{2}+f_{3}(\lambda, r) d x_{i} d x^{i}\right], \qquad i=1,2, \ldots D.\tag{6}$$

my doubt is that why the authors can utilize ricci flow in a lorentz space. can you help me

• And which equation in the article "Constructing Lifshitz spaces using the Ricci ﬂow" is in doubt? – Alex Trounev Jun 22 '19 at 3:17
• the (6), which is mentioned above, it is not a Remannian metric, why can use ricci flow – explorer Jun 22 '19 at 3:34
• Equation (6) is the definition of a metric. Equation (5) is the definition of the Hamilton-DeTurck Ricci ﬂow. This is not Ricci ﬂow as usual, but adapted for metric (6). And there is given the proof that there is a solution to the corresponding equations. – Alex Trounev Jun 22 '19 at 4:29
• but i think the author should prove the (5), because it is known that ricci flow is firstly proposed just for Remannian manifold not for lorentz and the ricci flow idea is derived from heat equation and energy funcational, but there is now such analogy in lorentz manifold – explorer Jun 22 '19 at 7:02
• Equations (9), (10) and the next equation from them (16) are of parabolic type. That agrees with the Ricci flow equation. Before equation (5) there is a link to the article D. M. DeTurck, J. Diﬀ. Geo. 18, 157-162 (1983). It is necessary to verify how DeTurck justified (5). – Alex Trounev Jun 22 '19 at 7:59