Characterizations Of Umbilically Synchronized Space-times

We study umbilically synchronized space-times $M$ that have vanishing electric part of the Weyl tensor and show that (i) If $dim(M)=4$, then $M$ is conformally flat, (ii) If $dim(M)>4$, $M$ is conformally flat if and only if the spatial slices are conformally flat. Next, we characterize umbilically synchronized spacetimes that are vacuum and have zero electric components of its Weyl tensor. Finally, we characterize conformal vector fields of an umbilically synchronized space-time.