Nontrivial exact fiber translation
= Nontrivial exact fiber translation
{title2=$T_{dh}^*\omega=\omega,\quad T_{dh}^*\alpha=\alpha+\pi^*dh$}
If $dh$ is not identically zero, translating cotangent fibers by $dh$ preserves the canonical <symplectic form> but changes the <Liouville one-form> by $\pi^*dh$. 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.