If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)?

The only thing that I have found is a theorem by Gromov which is conceptually very nice, but which seems to be very difficult to use. First, I should construct some embedding $i_0$ of $M$ - I guess that I could use Whitney here. Next, I should check that $i_0 ^* \Omega$ belongs to the same cohomology class as $\omega$. Finally, I should check that ${\rm d} i_0$ is homotopic through fiberwise-injective bundle maps $: TM \to T \Bbb R ^{2N}$ to some symplectic morphism. This does not look easy at all.

The thing with Gromov's theorem is that it's very general: it embeds in arbitrary symplectic manifolds (not just in $\Bbb R ^{2N}$), and it allows $\omega$ to have arbitrary non-constant rank - hypotheses that are too generous for me.

The closest thing to my needs is a corollary which says that if $\omega$ is an exact symplectic form, then $(M, \omega)$ embeds symplectically into $(\Bbb R ^{2N}, \Omega)$ provided that $\dim M \le N$. Unfortunately, the requirement that $\omega$ be exact is too strong for me.

Does anyone, then, know of more humane conditions (but valid for arbitrary symplectic manifolds) that guarantee what I want?

ADDENDUM: It would be useful too to find symplectic embeddings into some cotangent bundle - but I expect this to be even more complicated.

If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)?

The only thing that I have found is a theorem by Gromov which is conceptually very nice, but which seems to be very difficult to use. First, I should construct some embedding $i_0$ of $M$ - I guess that I could use Whitney here. Next, I should check that $i_0 ^* \Omega$ belongs to the same cohomology class as $\omega$. Finally, I should check that ${\rm d} i_0$ is homotopic through fiberwise-injective bundle maps $: TM \to T \Bbb R ^{2N}$ to some symplectic morphism. This does not look easy at all.

The thing with Gromov's theorem is that it's very general: it embeds in arbitrary symplectic manifolds (not just in $\Bbb R ^{2N}$), and it allows $\omega$ to have arbitrary non-constant rank - hypotheses that are too generous for me.

The closest thing to my needs is a corollary (corollary (a) on page 334 of Gromov's 1986 "Partial Differential Relations") which says that if $\omega$ is an exact symplectic form, then $(M, \omega)$ immerses symplectically into $(\Bbb R ^{2N}, \Omega)$ provided that $\dim M \le N$. Unfortunately, the requirement that $\omega$ be exact is too strong for me.

Does anyone, then, know of more humane conditions (but valid for arbitrary symplectic manifolds) that guarantee what I want?

ADDENDUM: It would be useful too to find symplectic embeddings into some cotangent bundle - but I expect this to be even more complicated.