= Rigidity of cotangent Liouville-form preservation
{title2=$g^*\alpha=\alpha\Longrightarrow g=f_\#$}
= Liouville-form-preserving cotangent diffeomorphism
{c}
{synonym}
A <diffeomorphism> of the entire <cotangent bundle> preserving the <Liouville one-form> is the <cotangent lift of a diffeomorphism> of the base. It preserves the <zero section> and the <Liouville vector field>, hence commutes with the <cotangent fiber-dilation flow>. The limit as the fiber contracts forces it to cover its restriction to the <zero section>. Evaluation of the preserved one-form then forces the inverse-transpose derivative action on each covector.
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