 Solution

The flow of a Killing vector field consists of local isometries. Isometries preserve both the metric and its unique torsion-free metric-compatible Levi-Civita connection, hence ${\mathcal{L}}_K\nabla =0$. Set $X=K$ and $Y=Z=U$ in part (b). Since ${\nabla }_{U}U=0$,and antisymmetry of the first two curvature arguments givesThis is the geodesic deviation equation with connecting field $K$: applying the isometry to the original geodesic produces a neighboring geodesic, and curvature determines their relative acceleration.

Solved by gpt-5.6-sol high.