How to solve this DE to get a matrix?

I have this DE

$$g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} = X_{i\bar{j}}~ \partial_t{f}^i (t) ~ \partial_t{f}^\bar{j} (t) ~~~ ~~~~~~~~~~~~(*)$$

Where m,n,p,r,i,j = 1,2,3 and the $$\bar{}$$ are just the complex coordinates. So $$g^{m\bar{n}}$$ and $$X_{i\bar{j}}$$ are 3x3 complex matrices.

I know the right hand side of the equation as a whole , i.e., I got $$X_{i\bar{j}}~ \partial_t{f}^i (t) ~ \partial_t{f}^\bar{j} (t)$$ by

sol = NDSolveValue[{l1[t] == 0, l3[t] == 0, l2[t] == 0, ic}, ff , {t, tmin,tmax}]

where l1[t], l2[t] and l3[t] are other DEs with initial conditions ic.

The question now can I solve (*) to get g ?

• How can you be sure that there is a unique solution? Actually, I doubt it, as this probably specifies only the squares Frobenius norm $|\partial_t g(t)|_{g(t)}^2$ of the Riemannian metric $\partial_t g(t)$ with respect to $g(t)$. Alas, within the space of Riemannian metrics that are conformal to the initial one (i.e., $g(t) = \exp(\lambda(t)) \, g(0)$, this might define a meaningful flow (depends somewhat on the right hand side). Maybe you have some context that you might want to share with us? Aug 8 '19 at 15:17
• If it solved by NDSolve , this says yeah there may by many solutions .. Aug 8 '19 at 15:35
• Why do you say $\partial_t g(t)$ is Riemannian ? I didn't mention that. It will mean it may has indices (-1,+,+,+) while no it's 3 x3 complex matrix . Aug 8 '19 at 15:40
• Okay, sorry. As a geometer, all matrix-valued functions called g are Riemannian or Lorentzian metrics. (Riemannian metrics are positive definite btw.). Thus, I expected this to be in some setting of Kähler geometry. Aug 8 '19 at 15:54