A note on Penrose limits

As a footnote to arXiv:1909.07756, I show that, given a (time-like) umbilic 3-surface $\mathrm{\Sigma }$ in a 4-dimensional space-time $M$, the Penrose limit taken along any null geodesic $\mathrm{\Gamma }$ which lies in $\mathrm{\Sigma }$ is a diagonalisable plane-wave.