Let (M,g) be an EInstein manifold, Ricci flat Ric(g)=0 and X a vector field, I consider M.R and the metric gX:
gX=g+(X∗+dt)⊗(X∗+dt)
The scalar curvature of gX is rX. The generalized Einstein equations are:
X=(drX)∗
Have we non trivial solutions of the generalized Einstein equations?