Let $\phi$ be a homogeneous symplectomorphism of tangent bundle $\dot{T}^*M=T^*M-0_M$ and let $\alpha_M$ be the canonical Liouville 1-form of $\dot{T}^*M$. Then is it true that $\phi^*\alpha_M=\alpha_M$?
Any homogeneous symplectomorphism of tangent bundle $\dot{T}^*M=T^*M-0_M$ preserves the canonical Liouville form?
SoYu
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