Here $(M,g)$, $(\tilde M,\tilde g)$ consists of a smooth manifold $M$ (resp. $\tilde M$) with Riemannian metric $g$ (resp. $\tilde g$).
Does the result above hold for pseudo-Riemannian manifolds as well? Or more specifically, say we have $M=\mathbb R^4$, and $g$ is the Lorentz metric induced by $$ \beta\colon\mathbb R^4\times\mathbb R^4\to\mathbb R\colon (x,y)\mapsto x_1y_1-x_2y_2-x_3y_3-x_4y_4. $$ Would Prop. 5.9 then hold still?