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Symplectomorphisms preserving tautological one-forms

  1. Jun 13, 2012 #1
    Symplectomorphisms preserving "tautological" one-forms

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

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

    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 [itex] p,q \in M [/itex] such that g(p) = q, so that [itex] g^* \alpha_q = \alpha_p [/itex]. Thus
    [tex] \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*} [/tex]
    That is, I have shown that [itex] -\omega_q(v_q,g_*v_p) = 0 [/itex]. 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 [itex] \omega [/itex] the one-form [itex] \alpha [/itex] can be recovered. Thoughts?
     
  2. jcsd
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