Well my main questions are colored in red.... Set of problem Let's suppose we have a metric gab(x) and the Killing vectors Ia ( Ia;k=0, gμαIμIα= 0) question Show that those killing vectors are the gradient of a Scalar field, and that it satisfies the equation IαRαβγρ=0 Show that the new metric bellow describes the same space. Gab= gab+IaIb Attempt to solve 1. Since Ia;k=0, Ik;a=0 we get the equations Ia,k - Γρak Iρ= 0 (1) Ik,a - Γρka Iρ= 0 (2) from (1)-(2) we get Ia,k-Ik,a=0 can I say now that the Iα(x) is the gradient of a scalar? I think I can. 2. IαRαβγρ=0 by using the commutation of covariant derivative: Ia;k;c-Ia;c;k=IdRdakc but we know that Ia;b=0 so 0;c - 0;k =IdRdakc IdRdakc=0 I think that this was an easy question. 3. I have that Gab= gab+IaIb how can I show that it describes the same space? one way I thought of is to calculate the R' riemann tensor of metric G and show that it is equal to R tensor of the metric g. But things get lost in calculations... what is your opinion?