I have a Riemannian metric on $R^3$ whose matrix is written as $h=P^TDP$, where $P\in SO(3)$ and $D$ is a diagonal matrix with positive valued smooth functions on the diagonal. Here $SO(3)$ is the special orthogonal space.
To be more clear at each point $p$, $h(p)$ in $T_pM$ is as $h(p)=(P^TDP)(p)$ with $P(p)\in SO(3)$ and $D(p)=diag(a(p),b(p),c(p))$. Also $P(p)=R_z({\gamma(p)})R_y({\beta(p)})R_x({\alpha(p)})$, where
$$R_x(\alpha(p))= \left(\begin{matrix} 1 & 0 & 0\\ 0 & \cos \alpha(p) & -\sin \alpha(p)\\ 0 & \sin \alpha(p) & \cos \alpha(p) \end{matrix}\right), R_y(\beta(p))= \left(\begin{matrix} \cos \beta(p) & 0&\sin \beta(p) \\ 0 & 1 & 0\\ -\sin \beta(p) & 0&\cos \beta(p) \end{matrix}\right),$$ $$ \begin{align*} R_z(\gamma(p))= \left(\begin{matrix} \cos \gamma(p) & -\sin \gamma(p) & 0\\ \sin \gamma(p) & \cos \gamma(p) & 0\\ 0 & 0 & 1 \end{matrix}\right). \end{align*}$$
If I am not wrong $D$ is also a Riemannain metric! (Yes?)
Now my question is this:
There is any relation between the geodesics of $h$ with those of $D$?
Many thanks in advance!