Plotting matrices

I have this expression:

$$\int g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} ~ d^6 y = f(t)$$

Where m,n,p,r=1,2,3 and the $$\bar{}$$ are just the complex coordinates.

Any idea guys how to plot the left hand side of g with the knowledge of f(t) ?

$$g^{m\bar{n}}$$ is a 3x3 complex matrix.

It’s like $$g_{\mu\nu}$$ in the Riemannien line element $$ds^2 =g_{\mu\nu} dx^\mu dx^\nu$$ , but we know the elements of $$g_{\mu\nu}$$, while $$g_{mr}$$ only what I know about is the above relation to some function f(t). I can plot this in 3D or ContourPlot, …

Maybe I search for a final result like in answer 1 in the following question:

How would I plot the metric $ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2$ in Mathematica?

Any help appreciated!