General Relativity, identity isotropic, Ricci tensor

1. Mar 28, 2017

binbagsss

1. The problem statement, all variables and given/known data

Attached

2. Relevant equations

3. The attempt at a solution

So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold?

I don't quite understand by what is meant by 'these vectors give preferred directions'. Can someone explain this more please? How do the eigenvectors indicate a preferred direction?

Also, if it is only isotropic about a single point, then at all other points we do expect there to be preferred directions? So don't we expect something like $R^u_v$ evaluated at the isotropic point would specify no preferred directions, and so indeed $R^u_v=c\delta^u_v$ is needed, however at all the other points space is not isotropic, these preferred directions can be manifest?

What point is $R^u_v$ being evaluated at?