A $2n$-dimensional manifold $M$ is said to be *almost symplectic* if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \subset F(M)$ of the frame bundle where $G < GL(2n,\mathbb{R})$ is isomorphic to the symplectic group $Sp(2n,\mathbb{R})$.


The intrinsic torsion of such a $G$-structure is captured by the three-form $d\omega \in \Omega^3(M)$. The bundle $\wedge^3 T^\*M$ breaks up into the Whitney sum of two $G$-stable sub-bundles corresponding to the $\omega$-traceless 3-forms and their $\omega$-perpendicular complement. This therefore gives rise to four types of almost symplectic manifolds:


* **symplectic**, where $d\omega = 0$
* **locally conformal symplectic**, where $d\omega = \omega \wedge \varphi$ for some one-form $\varphi$ which is closed and hence $\varphi = df$ locally, allowing us to construct a local symplectic form $e^{-f}\omega$.
* **name?**, where the volume form $\omega^n$ is left invariant by the *hamiltonian vector fields* $X\_f = \omega^\sharp(df)$
* **generic**, where $d\omega$ is none of the above.


My question is whether there is an accepted name for the third type. I would also appreciate a link to where this classification was made explicit for the first time.


Thank you.


**Edit** As Robert Bryant pointed out below, the condition **name?** is actually $d\omega^{n-1} = 0$. I will leave the question unmodified, except for this.



Name for a class of almost symplectic manifolds
I'm a bit confused by your question, because I believe that, if one defines an $\omega$-Hamiltonian vector field to be a vector field of the form $X\_f = \omega^\#(\mathrm{d}f)$ where $f$ is a (smooth) function on $M$, then $\omega^{n}$ is *always* invariant under the flow of $X\_f$.


To see this, recall that, when $n>1$, if $\omega$ is a non-degenerate $2$-form on $M^{2n}$, its exterior derivative can be written uniquely in the form
$$
\mathrm{d}\omega = \phi\wedge\omega + \psi
$$
where $\phi$ is a $1$-form and $\psi\in\Omega^3(M)$ is *$\omega$-primitive*, i.e., $\omega^{n-2}\wedge\psi = 0$.


Meanwhile, by Cartan's formula for the Lie derivative with respect to $X\_f$, we have, since $\iota(X\_f)\omega = -\mathrm{d}f $ (where $\iota(X)$ denotes interior product with $X$),
$$
\begin{align}
\mathcal{L}\_{X\_f}\omega^n &= n\,\omega^{n-1}\wedge \mathcal{L}\_{X\_f}\omega
= n\,\omega^{n-1}\wedge\bigl(\iota(X\_f)(\mathrm{d}\omega) + \mathrm{d}(\iota(X\_f)\omega)\bigr)\\
&=n\,\omega^{n-1}\wedge\bigl(\iota(X\_f)(\phi\wedge\omega + \psi) + \mathrm{d}(-\mathrm{d}f))\bigr)\\
&=n\,\omega^{n-1}\wedge\bigl(\phi(X\_f)\wedge\omega + \phi\wedge\mathrm{d}f+ \iota(X\_f)\,\psi)\bigr)\\
&= n\,\omega^{n-1}\wedge\bigl(\iota(X\_f)\,\psi\bigr) =0
\end{align}
$$
since $\omega^{n-2}\wedge\psi=0$ implies


$$
0 = \iota(X\_f)(\omega^{n-1}\wedge\psi) = (n{-}1)\,\omega^{n-2}\wedge(-\mathrm{d}f)\wedge\psi + \omega^{n-1}\wedge\bigl(\iota(X\_f)\,\psi\bigr).
$$


Instead of the 'Hamiltonian flow invariance' criterion you propose for **name?** (by which, I think, you are trying to capture the condition $\phi=0$), you should instead just require that $\omega^{n-1}$ be *closed*. (This condition is sometimes known as 'balanced' in the literature.)


By the way, your second type "locally conformally symplectic" is only apt if $n>2$. When $n=2$, you do *not* automatically get that $\phi$ is closed from the condition $\mathrm{d}\omega = \phi\wedge\omega$. (In fact, it is generically not true, even though $\psi$ does vanish identically when $n=2$.)


