Name for a class of almost symplectic manifolds

Question

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.

Answer 1

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 $n=2$.)