Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 15 / 4 / a

For a covector $p\in T_x^{\ast }X$, define the cotangent lift of a diffeomorphism byThe inverse derivative here maps $T_{f(x)}X$ to $T_{x}X$. This is a smooth diffeomorphism, its inverse is $(f^{-1}{)}_{\#}$, and it satisfies $\pi \circ f_{\#}=f\circ \pi$.

For $v\in T_{(x,p)}{T}^{\ast }X$, the pullback of a differential form and the defining formula for the Liouville one-form giveHence $f_{\#}^{\ast }\alpha =\alpha$, and applying the exterior derivative also proves $f_{\#}^{\ast }\omega =\omega$. The cotangent lift of a diffeomorphism thus preserves both canonical forms, with no extra choice of metric or connection.