Tautological one-form

Summary

The tautological one-form is a canonical $1$-form on the cotangent bundle of any smooth manifold. In coordinates $(q, p)$ it is $\sum_i p_i \, dq^i$, and it needs no choice of coordinates or metric. Its exterior derivative gives the canonical symplectic form of the cotangent bundle, which makes every cotangent bundle a symplectic manifold. It is also called the Liouville form or the canonical one-form.