A differential geometry question from continuum mechanics

[wrobel]

Consider the flow of a continuous medium with a smooth velocity field $\boldsymbol v(x)$ in $\mathbb{R}^3\ni x$. The density of the medium is $\rho(x)$. Here $\mathbb{R}^3$ is equipped with the standard inner product $\delta_{ij}$.

By definition, the momentum of a material volume $D$ (where $D$ is a bounded domain in $\mathbb{R}^3$) is given by the formula:
$$\boldsymbol P=\int_D\boldsymbol v\mu, \qquad(1)$$
where $\mu=\rho\sqrt g dx^1\wedge dx^2\wedge dx^3$ is a differential form (the infinitesimal mass).

And what about formula (1) if we replace $\mathbb{R}^3$ with some other Riemannian manifold with non-zero curvature?
In this case, formula (1) becomes senseless, and I have no idea how to fix it—or if it is even possible.
Any opinions on this?

 

[QuarkyMeson]

On a curved manifold, velocities at different points belong to different tangent spaces, so they cannot be added directly. You need a chosen way to transport vectors or a symmetry of the manifold to define a total momentum.

So I don't think it's generally possible, but you might be able to rescue (1) in some cases.

Side note, even when a manifold has no symmetries and momentum is lost, ideal flows still have conserved quantities of a different kind, they're just now topological ones. This would be bleed into some of the braiding stuff I've been reading, but it's very tangential to this.

 

[Matterwave]

And what about formula (1) if we replace $\mathbb{R}^3$ with some other Riemannian manifold with non-zero curvature?
In this case, formula (1) becomes senseless, and I have no idea how to fix it—or if it is even possible.
Any opinions on this?

I am not aware of a way to in general fix this integral in the sense that there is a unique 1:1 correspondence between this integral and a "fixed version" that works in general Riemannian manifolds. And so, I might not have too many intelligent things to say about this. So take the following with a huge grain of salt.

One thing I might try, could be to turn the vector into a 1-form using the metric $g(v,\,\,)$ and then contracting with some Killing Field $\xi$ to perhaps obtain some conserved aggregate quantity. You'd get an integral like:

$$ I(\xi) = \int_D g(v, \xi)\mu$$

Your form of $\mu$ seems to me to work on a general manifold. This integral that I constructed is going to come out with just number though and not a vector.

Trying this integral from a transport-related view seems gnarly to me since the transport would be path dependent.