A question for expert in geometrical method and Riemannian metrics I'm a physical oceanographer with great interest in Theoretical Geophysical Fluid Dynamics. I have some ideas on the possibility to derive the so-called: geostrophic equilibrium (i.e. on a rotating planet the wind is flowing along the isobars due to the combined effect of the pressure gradient and the Coriolis force) through a suitable metric on a rotating sphere (or something similar to the parallel transport. The naive idea is that it might be possible for a particle to follow the trajectory imposed by the Coriolis force like the geodetic path on a gravitational field in general relativity. Unfortunately, I am not an expert in geometric and metric theory on a manifold. If you are interested in the problem I would be very happy.  Notice that the standard derivation of the geostrophic balance is based on scaling arguments, I Think that some geometrical properties may be intrinsically connected to the balance.
 A: I believe that your intuition might be sort of correct. However, you might be delighted (or not) to hear that the story is possibly a bit more-interesting. The motion of a particle moving freely (meaning only subject to the fictitious forces of the non-inertial, uniformly rotating reference frame) on any surface of revolution, that rotates uniformly around its axis of rotational symmetry, is the projection of a geodesic on a circle bundle over the said surface. In other words, it is the projection of a geodesic of circle-action invariant metric on a 3D circle bundle over the surface. As such, this reminds me more of a classical version not of the standard 4D general relativity theory but of the 5D Kaluza-Klein theory that unifies gravity with electromagnetism.
Assume you have a surface of revolution, with the $z-$axis being the axis of rotational symmetry of the surface. Usually, such a surface is obtain by taking a planar curve in the $x, z$ coordinate plane, say parametrized as
\begin{align}
x \, &= \, f_1(q)\\
z \, &= \, f_2(q)\\
\end{align}
and rotating it around the $z-$axis, which yields the parametrization $f \, : \, (a, b) \times [0, 2\pi) \, \to \, \mathbb{R}^3$
\begin{align}
& x \, = \, f_1(q) \cos(\lambda)\\
f \, :\,\,\,& y \, = \, f_1(q) \sin(\lambda)\\
& z \, = \, f_2(q)\\
\end{align}
where $(q, \, \lambda)$ are the curvilinear 2D coordinates on the surface. In manifold theory language, this is a 2D chart of the surface. The corresponding metric in this surface chart is then the pull-back of the standard 3D Euclidean metric, which can be written as
$$ds^2 \,=\, g_{11}(q) \, dq^2 \,+\, g_{22}(q) \, d\lambda^2$$ where
$$g_{11}(q) \,=\, f_1'(q)^2 +\, f_2'(q)^2$$
$$g_{22}(q) \,=\, f_1(q)^2$$
As you can see, the metric does not depend on $\lambda$ explicitly, but only on $d\lambda$, which means that it is invariant with respect to all transformations of type $(q, \, \lambda) \mapsto (q, \, \lambda + \theta)$ for any parameter $\theta$. Here $\theta$ is a rotation angle around the $z-axis$ and as such it is a variable on a circle. This invariance of the metric is a direct consequence of its rotational symmetry. But now, consider the following metric on the 3D manifold
$$(a, b) \times [0, 2\pi) \times [0, 2\pi)$$
$$d\sigma^2 \,=\, g_{11}(q) \, dq^2 +\, g_{22}(q) \, (\,d\lambda + d\theta\,)^2 \,+\, d\theta^2$$
This metric gives rise to the Lagraingian
$$\mathcal{L} \,=\, \frac{1}{2}\, g_{11}(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)^2 \,+\, \frac{1}{2}\,\left(\frac{d\theta}{dt}\,\right)^2$$
The geodesic equations of the 3D metric (time is an affine parameter for the metric) can be written as the Euler-Lagrange equations of this Lagrangian, which are
\begin{align}
&\frac{d}{dt} \left(\,g_{11}(q) \left(\frac{dq}{dt}\right)\,\right) \,=\,  \frac{1}{2}\, g_{11}'(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}'(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)^2\\
&\\
&\frac{d}{dt} \left(\,g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)\,\right) \,=\,0\\
&\\
&\frac{d^2\theta}{dt^2}\,+\, \frac{d}{dt} \left(\,g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)\,\right) \,=\,0
\end{align}
However, the last two equations simplify to
\begin{align}
&\frac{d}{dt} \left(\,g_{11}(q) \left(\frac{dq}{dt}\right)\,\right) \,=\,  \frac{1}{2}\, g_{11}'(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}'(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)^2\\
&\\
&\frac{d}{dt} \left(\,g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)\,\right) \,=\,0\\
&\\
&\frac{d^2\theta}{dt^2} \,=\,0
\end{align}
and after their integration once,
\begin{align}
&\frac{d}{dt} \left(\,g_{11}(q) \left(\frac{dq}{dt}\right)\,\right) \,=\,  \frac{1}{2}\, g_{11}'(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}'(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right)^2\\
&\\
&g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \frac{d\theta}{dt}\,\right) \,=\,h\\
&\\
&\frac{d\theta}{dt} \,=\,\omega
\end{align}
where $h$ is a constant, $\omega$ is also a constant which turns out to be the magnitude of the constant angular velocity of the uniform rotation of the surface. Thus,
\begin{align}
&\frac{d}{dt} \left(\,g_{11}(q) \left(\frac{dq}{dt}\right)\,\right) \,=\,  \frac{1}{2}\, g_{11}'(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}'(q) \, \left(\, \frac{d\lambda}{dt} + \omega\,\right)^2\\
&\\
&\frac{d\lambda}{dt}\,=\,\frac{h}{g_{22}(q)} \, - \, \omega\\
&\\
&\frac{d\theta}{dt} \,=\,\omega
\end{align}
Now, if you compare these equations to the equations of motion of the standard classical Lagrangian of a freely moving particle in on a uniformly rotating surface of revolution, with constant angular velocity of magnitude $\omega$, the latter Lagrangian is
$$\mathcal{L} \, =\, \frac{1}{2}\, g_{11}(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}(q) \, \left(\, \frac{d\lambda}{dt} + \omega\,\right)^2 $$ and its Euler-Lagrange equations of motion are
\begin{align}
&\frac{d}{dt} \left(\,g_{11}(q) \left(\frac{dq}{dt}\right)\,\right) \,=\,  \frac{1}{2}\, g_{11}'(q) \, \left(\frac{dq}{dt}\right)^2 +\, \frac{1}{2}\,g_{22}'(q) \, \left(\, \frac{d\lambda}{dt} + \omega\,\right)^2\\
&\\
&\frac{d\lambda}{dt}\,=\,\frac{h}{g_{22}(q)} \, - \, \omega
\end{align}
I suspect that you will probably have to extend your fluid dynamics model from the ellipsoid of revolution (the geodetic model of the Earth) to a fluid dynamics model on its 3D circle bundle, where the material derivative is going to be exactly the Levi-Civita covariant differentiation on this circle bundle and the the velocity field of the fluid on this circle bundle should be invariant with respect to the circle-action of the fiber, i.e. the velocity field probably should not depend on $\theta$.
