Tautological one-form

Remarks

In local coordinates $q^1, \ldots, q^n$ on $Q$ with the induced coordinates $p_1, \ldots, p_n$ on the fibres, $\theta = \sum_i p_i \, dq^i$. The form $\omega = -d\theta = \sum_i dq^i \wedge dp_i$, with $d$ the exterior derivative, is the canonical symplectic form of $T^*Q$; some authors use the opposite sign.

Characterization: for every $1$-form $\alpha$ on $Q$, viewed as a section $\alpha \colon Q \to T^*Q$, one has $\alpha^* \theta = \alpha$, and $\theta$ is the only $1$-form on $T^*Q$ with this property.

Example: for $Q = \R$ with coordinate $q$ and fibre coordinate $p$, $\theta = p\, dq$; for $\alpha = c\, dq$ the section is $q \mapsto (q, c)$ and $\alpha^*(p\, dq) = c\, dq = \alpha$.

Non-example: the form $q\, dp$ on $T^*\R$ is not the tautological form. For the section $\alpha = dq$, that is $q \mapsto (q, 1)$, the pullback of $q\, dp$ is $q \cdot d(1) = 0 \ne dq$.