# Symplectomorphisms preserving tautological one-forms

1. Jun 13, 2012

### Kreizhn

Symplectomorphisms preserving "tautological" one-forms

1. The problem statement, all variables and given/known data

Let $(M,\omega)$ be a symplectic manifold such that there is a smooth one-form $\alpha \in \Omega^1(M)$ satisfying $\alpha = -d\omega$. Let $v \in \Gamma(TM)$ be the unique vector field such that $\iota_v \omega = -\alpha$. If $g: M \to M$ is any symplectomorphism that preserves $\alpha$ (that is $g^*\alpha = \alpha$) show that $g_* v = v$.

3. The attempt at a solution

This has been giving me trouble for a few days and I don't think it should be that difficult. The attempt I like most thus far is the following: Let $p,q \in M$ such that g(p) = q, so that $g^* \alpha_q = \alpha_p$. Thus
\begin{align*} \alpha_q(g_* v) &= (g^*\alpha_q) v_p = \alpha_p(v_p) = -\omega_p(v_p,v_p) = 0 \\ &= -\omega_q(v_q, g_*v_p). \end{align*}
That is, I have shown that $-\omega_q(v_q,g_*v_p) = 0$. Now a priori there is no reason to suspect that they must be equal, but I feel that this implication may be the key if we combine the fact that v is the unique vector field against whose retraction with $\omega$ the one-form $\alpha$ can be recovered. Thoughts?