Nontrivial exact fiber translation ($T_{dh}^{\ast }\omega =\omega ,T_{dh}^{\ast }\alpha =\alpha +{\pi }^{\ast }dh$)

If $dh$ is not identically zero, translating cotangent fibers by $dh$ preserves the canonical symplectic form but changes the Liouville one-form by ${\pi }^{\ast }dh$. Any positive-dimensional smooth manifold has a suitable nonconstant bump function in a chart. In dimension zero every differential one-form is zero, so this distinction cannot occur.