I'm hoping that the following are true. In fact, they are probably easy, but I'm not seeing the answers immediately.


Let $M$ be a smooth $m$-dimensional manifold with chosen positive smooth density $\mu$, i.e. a chosen (adjectives) volume form. (A *density* on $M$ is a section of a certain trivial line bundle. In local coordinates, the line bundle is given by the transition maps $\tilde\mu = \left| \det \frac{\partial \tilde x}{\partial x} \right| \mu$. When $M$ is oriented, this bundle can be identified with the top exterior power of the cotangent bundle.) **Hope 1: Near each point in $M$ there exist local coordinates $x: U \to \mathbb R^m$ so that $\mu$ pushes forward to the canonical volume form $dx$ on $\mathbb R^m$.**


Hope 1 is certainly true for volume forms that arise as top powers of symplectic forms, for example, by always working in Darboux coordinates. If Hope 1 is true, then $M$ has an atlas in which all transition maps are volume-preserving. My second Hope tries to describe these coordinate-changes more carefully.


Let $U$ be a domain in $\mathbb R^m$. Recall that a change-of-coordinates $\tilde x(x): U \to \mathbb R^m$ is *oriented-volume-preserving* iff $\frac{\partial \tilde x}{\partial x}$ is a section of a trivial ${\rm SL}(n)$ bundle on $U$. An *infinitesimal change-of-coordinates* is a vector field $v$ on $U$, thought of as the map $x \mapsto x + \epsilon v(x)$. An infinitesimal change-of-coordinates is necessarily orientation-preserving; it is volume preserving iff $\frac{\partial v}{\partial x}(x)$ is a section of a trivial $\mathfrak{sl}(n)$ bundle on $U$. **Hope 2: The space of oriented-volume-preserving changes-of-coordinates is generated by the infinitesimal volume-preserving changes-of-coordinates, analogous to the way a finite-dimensional connected Lie group is generated by its Lie algebra.**


Hope 2 is not particularly well-written, so Hope 2.1 is that someone will clarify the statement. Presumably the most precise statement uses infinite-dimensional Lie groupoids. The point is to show that a certain *a priori* coordinate-dependent construction in fact depends only on the volume form by showing that the infinitesimal changes of coordinates preserve the construction.


***Edit:*** I have preciseified Hope 2 as [this question](https://mathoverflow.net/questions/7853/is-the-space-of-volume-preserving-maps-path-connected).



Normal coordinates for a manifold with volume form
Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in my paper *[A volume-preserving counterexample to the Seifert conjecture](https://arxiv.org/abs/math/9504230)* (Comment. Math. Helv. 71 (1996), no. 1, 70-97), where I also established a similar result for the volume-preserving PL pseudogroup. In the PL case, the corresponding decoration on the manifold is a piecewise constant volume form.


In my opinion, the most exciting result on this theme is the Ulam-Oxtoby theorem. (But Moser's version is the most useful one and the most elegant.) The theorem is that if you have a topological manifold with any Borel measure that has no atoms and no bald spots, then it is modeled on the pseudogroup of volume-preseserving homeomorphisms. For example, you can start with Lebesgue measure in the plane and add uniform measure on a circle, and there is a homeomorphism that takes that measure to Lebesgue measure.


For a long time I have wondered about the pseudogroup of volume-preserving Lipschitz maps. The question is whether there is a corresponding cone of measures, and if so, how to characterize it.


