# 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.

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