Normal coordinates for a manifold with volume form 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.
 A: 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 (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.
