# Sum of Killing vector fields is a Killing vector field

Let $(M,g)$ be a Riemannian manifold. A smooth vector field $X$ is called a Killing vector field if the flow of $X$ acts by isometries, or, equivalently, if $L_X g = 0$. Now why is the sum of Killing vector fields a Killing vector field?

• What is the rule for sums in Lie derivative definition? – DiegoMath Jan 21 '15 at 21:39
• If you want, you can use the following definition instead: $$g(\nabla_YX,Z)+g(Y,\nabla_ZX)=0,\ \forall Y,Z\in\Gamma(TM).$$ – DiegoMath Jan 21 '15 at 21:47
• @DiegoMath The problem is I don't really know what the flow of a sum of two vector fields is. Is $L_{X+Y}g = L_x g + L_Y g$ true? I also don't see why. – Balerion_the_black Jan 21 '15 at 22:09
• Use this computation for Lie derivatives:$$(L_Xg)(Y,Z)=X(g(Y,Z))-g(L_XY,Z)-g(Y,L_XZ).$$ Can you conlude the statement? Remember that $L_XY=[X,Y]$. – DiegoMath Jan 21 '15 at 22:18

Recall that $\mathcal{X}(M)$, the space of vector fields on $M$, is the Lie algebra of the Lie group $\mathrm{diff}(M)$, the group of automorphisms of $M$. The space $\mathcal{K}(M)\subset\mathcal{X}(M)$, which consists of the Killing vector fields, is the Lie algebra of the group of isometries on $M$. Hence, $\mathcal{K}(M)$ is a vector space.