Natural tensors under Killing isometries

By the equivalent covariant characterization of a Killing field in Killing fields of a Lorentzian metric, and metric compatibility and torsion-freeness, $({\mathcal{L}}_{\xi }g)(X,Y)=g({\mathrm{\nabla }}_X\xi ,Y)+g(X,{\mathrm{\nabla }}_Y\xi )$: expand the Lie derivative from its flow definition, use $[\xi ,X]={\mathrm{\nabla }}_{\xi }X-{\mathrm{\nabla }}_X\xi$, and cancel the two ${\mathrm{\nabla }}_{\xi }$ terms with metric compatibility. Thus ${\mathrm{\nabla }}_{(a}{\xi }_{b)}=0$ implies ${\mathcal{L}}_{\xi }g=0$. The local flow identity ${\mathrm{\Phi }}_{s+h}={\mathrm{\Phi }}_h\circ {\mathrm{\Phi }}_s$ and the pullback order give $\frac{d}{ds}{\mathrm{\Phi }}_s^{\ast }g={\mathrm{\Phi }}_s^{\ast }({\mathcal{L}}_{\xi }g)=0.$ In local tensor components [F5] makes each component constant in $s$; since ${\mathrm{\Phi }}_0$ is the identity, ${\mathrm{\Phi }}_s^{\ast }g=g$. Hence ${\mathrm{\Phi }}_s$ is a local isometry.

Define the pulled-back connection by $({\mathrm{\Phi }}_s^{\ast }\mathrm{\nabla }{)}_{X}Y:={\mathrm{\Phi }}_{s\ast }^{-1}({\mathrm{\nabla }}_{{\mathrm{\Phi }}_{s\ast }X}{\mathrm{\Phi }}_{s\ast }Y)$. A diffeomorphism preserves Lie brackets, so its pullback connection has zero torsion whenever $\mathrm{\nabla }$ does. Since ${\mathrm{\Phi }}_s^{\ast }g=g$, metric compatibility of $\mathrm{\nabla }$ also makes ${\mathrm{\Phi }}_s^{\ast }\mathrm{\nabla }$ compatible with $g$. The uniqueness in [F3] gives ${\mathrm{\Phi }}_s^{\ast }\mathrm{\nabla }=\mathrm{\nabla }$. The flow composition law also gives ${\mathrm{\Phi }}_{s\ast }\xi =\xi$.

Substituting ${\mathrm{\Phi }}_s^{\ast }\mathrm{\nabla }=\mathrm{\nabla }$ and ${\mathrm{\Phi }}_{s\ast }\xi =\xi$ into the curvature definition [F4] gives ${\mathrm{\Phi }}_s^{\ast }R=R$. Pullback commutes with tensor products and contractions by their slot definitions, and the preserved connection makes it commute with each covariant derivative. Repeating these identities proves preservation of every finite covariant derivative and contraction stated above; composing preserved scalar components with a smooth function preserves the resulting scalar invariant.