Tautological one-form on a cotangent bundle

Let $M$ be a smooth manifold and $\pi:T^*M\to M$ its cotangent bundle. The tautological one-form $\theta$ is the differential one-form on $T^*M$ defined at a covector $\alpha_x\in T_x^*M$ by

\theta_{\alpha_x}(v)=\alpha_x\!\left(d\pi_{\alpha_x}(v)\right), \qquad v\in T_{\alpha_x}(T^*M).

Thus $\theta$ evaluates the covector represented by the base point of $T^*M$ on the component of $v$ projected to $M$ . The construction uses only the cotangent projection and therefore requires no coordinates, metric, or connection.

Coordinate expression

For local coordinates $q^1,\ldots,q^n$ on $M$ and induced fiber coordinates $p_1,\ldots,p_n$ on $T^*M$ ,

\theta=\sum_{i=1}^n p_i\,dq^i.

Although this formula is coordinate-dependent in appearance, the defining evaluation formula proves that the one-form is intrinsic.

Canonical symplectic form

With the convention used here, the canonical symplectic form on $T^*M$ is

\omega_{\mathrm{can}}=-d\theta=\sum_{i=1}^n dq^i\wedge dp_i.

It is closed because $d^2=0$ , and the coordinate expression shows that it is nondegenerate. Hence every cotangent bundle carries canonical symplectic geometry without any auxiliary choice.

Conventions and scope

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.2, the tautological form and canonical symplectic form on a cotangent bundle.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, 1978. CaltechAUTHORS record. Relevant: cotangent bundles and canonical forms.