What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-Laplace Operator, $g_{ij}$ the metric tensor).

With $\Delta g_{ij}=0$, I mean the Laplace-Beltrami operator, applied componentwise to the components of the metric tensor.

Thank you in advance!

nointerest, it's just of specialized interest. There was a time when Liouville metrics were of great interest indeed, but now they are mainly of interest to the integrable systems folks, not so much in general relativity. – Robert Bryant Jul 4 '16 at 14:02