Connection compatible with a volume form

Question

Let $M$ be a smooth, orientable $n$-manifold and $\eta$ a volume form on $M$. Does there exist a connection $A$ on $TM$ such that $$\tag{$*$}D\eta=0,$$ where $D$ is the appropriate covariant derivative associated to $A$ on $\Omega^n(M)$? Can one assume $A$ to be symmetric?

I have a feeling that if one writes $(*)$ plus the torsion equation and then applies the Frobenius theorem, some local result can be obtained. But that doesn't give global conditions (on $M$ nor $\eta$). A reasonable assumption would be $M$ parallelizable.

Answer 1

If $\eta$ vanishes somewhere, then the only way it can be parallel (for any connection at all) is if it vanishes identically. Thus I will assume $\eta$ is non-vanishing.

Let $g$ be a Riemannian metric on $M$ and $\omega$ the volume form of $g.$ Since $\omega,\eta$ are smooth non-vanishing sections of a line bundle, there is some positive function $f \in C^\infty(M)$ such that $\eta = f \omega.$ (I'm assuming here that we take the orientation of $\omega$ to match that of $\eta.$)

The conformally related metric $\tilde g = f^{2/n}g$ then has volume form $\tilde \omega = (f^{2/n})^{n/2} \omega = \eta,$ which is made parallel by the Levi-Civita connection of $\tilde g.$ Thus there are no conditions necessary - there is always a symmetric connection that does the job.

Answer 2

An important observation to start with is that both properties of being torsion-free and being compatible with a volume form are preserved by taking affine combinations. That is, if the connections $\nabla^1,\ldots,\nabla^k$ have these properties, then so does the average connection $$\overline{\nabla}=\sum a_i\nabla^i,\quad \sum a_i=1.$$

Another important fact is that every volume form is integrable: for every $p\in M$ there is a coordinate chart $(U,\mathbf{X})$ centered at $p$, with respect to which we have $$\eta=dx^1\wedge\ldots\wedge dx^n.$$ The trivial connection on $U$ corresponding to $\mathbf{X}$ is both torsion-free and compatible with $\eta$.

Finally, one only needs to glue all these local connections using a partition of unity to obtain a global torsion-free connection which makes $\eta$ parallel.

Note that there are many choices to be made on the way (the trivializing coordinate charts are far from being canonical, as is the partition of unity). This implies that there are many different connections with the desired properties, as opposed to the Riemannian-metric case, where such a connection is unique by the Levi-Civita theorem.