0

Let $V$ be a submanifold of the cotangent bundle $T^*X$ of a smooth manifold $X$. Then we can consider the vector bundle $(TV)^{\bot}$ whose fiber is the symplectic orthogonal complement of a tangent space of $V$.

In some references, there is the statement: the vector bundle $(TV)^{\bot}$ is generated by Hamiltonian vector fields $X_f$ where $f$ vanishes on $V$.

I can check this in some particular cases. But I cannot prove this in general. If you know about that, please teach me.

SoYu
  • 307
  • If $f$ vanishes on $V$, then $\omega(X_f,w)={\rm d}f(w)=0$ for all $w$ tangent to $V$, which means that $X_f$ is normal to $V$ whenever $f$ vanishes at $V$. Can you go on with this? – Ivo Terek Nov 25 '19 at 14:34
  • @IvoTerek Thanks a lot. But I don't know how to express any element in $(TV)^{\bot}$ as a finite sum of such hamiltonian vector fields. – SoYu Nov 29 '19 at 07:29

0 Answers0