Is the Metric g a Complex Manifold?

[cuallito]

Hi so I was just wondering if the metric $g=diag(-e^{iat},e^{ibx},e^{icy})$ (where a,b,c are free parameters and t,x,y are coordinates) corresponds to a complex manifold (or is nonsensical), and what the manifold looks like?

 

[ChrisVer]

what do you mean by what the manifold looks like?
well you can always compute the necessary quantities (like Riemman curvature) and see what that is. I think there exists programs in mathematica that do the job...