# conformal changes to Lorentzian curvature

Let $$(M,g)$$ be a Lorentzian manifold and let $$R$$ be the curvature tensor. We say $$R\leq 0$$ if $$g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$

My question is whether given a Lorentzian manifold $$(M,g)$$, it is always possible to find a metric $$\hat{g}=cg$$ such that the curvature of $$(M,\hat{g})$$ is non-positive.

• I suppose you want $X,Y$ to be sections, not elements, of $TM$. What is $c$? Jun 3, 2020 at 12:24