We're looking at some compact Riemannian (maybe pseudo-Riemannian) manifold ($M$) without boundary and metric $g$. How would I go about finding the possible Weyl rescalings $w(x^{\mu}$) of the metric:
$$\tilde{g}=wg$$
such that the volume of the manifold is unchanged? In other words:
$$V=\intop_{M}\sqrt{|g|}d^{4}x=\intop_{M}\sqrt{|wg|}d^{4}x$$
Note I am NOT looking for isometries of the metric (which makes this question different from the one here), the integrands are NOT equal, but their integral is (I'm not lookingto preserve differential volume). It's easy to come up with some simple examples right “off the bat”. For example, consider the 1d “volume” of the unit circle:
$$V=\intop_{0}^{2\pi}1d\theta=2\pi$$
Where our “metric” is just $1$. Clearly a Weyl rescaling $$w=Sin^{2}(\theta)+\frac{1}{2}$$ will preserve the volume (I don't believe the volume form can be zero at any point hence adding the half). How can I find solutions to this in general? I'd like to work with higher dimensional spaces where I can't just “eyeball” a solution.
There must be a differential equation of some type which the $w$ has to satisfy correct?
EDIT I believe the answer might be the special affine group, except that I'm not interested in preserving the volume form necessarily but rather the total volume. So maybe the special affine group on the inner product space?
