# Comparing PDE solutions for different Riemannian metrics

I'm looking for the approach to compare PDE solutions on the Remannian manifolds when those solutions are obtained under two different metrics. To be more specific, suppose we have two Riemannian manifolds $(M,g_{1})$ and $(M,g_{2})$, partial differential operator $D$ is dependent on the metric. Then is there any way (I'm not having in mind any specific way) to compare solutions $u$, o the equation $Du=0$?

I know it is rather vague question. I don't even know in what way those solutions to compare. Any advice, link to some literature would be highly appreciated.

Lets assume you have a diffeomorphism $\phi:M_1 \mapsto M_2$ then I would map the solution on $M_2$ back to $M_1$ via the inverse mapping $\phi^{-1}$ and compare the two solutions.
• But then, how would you choose that diffeomorphism? In addition, I'm not considering different manifolds $M_1$ and $M_2$. It is the same manifold but the metrics are different. Aug 28, 2014 at 16:05
• I get your point. How do you define your metrics $g_1$ and $g_2$ ? Are they defined by some parametrization of the manifold?