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Cotangent bundle
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=== The tautological one-form === {{main|Tautological one-form}} The cotangent bundle carries a canonical one-form θ also known as the [[symplectic potential]], ''Poincaré'' ''1''-form, or ''Liouville'' ''1''-form. This means that if we regard ''T''*''M'' as a manifold in its own right, there is a canonical [[Section (fiber bundle)|section]] of the vector bundle ''T''*(''T''*''M'') over ''T''*''M''. This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that ''x''<sup>''i''</sup> are local coordinates on the base manifold ''M''. In terms of these base coordinates, there are fibre coordinates ''p''<sub>''i''</sub> : a one-form at a particular point of ''T''*''M'' has the form ''p''<sub>''i''</sub> ''dx''<sup>''i''</sup> ([[Einstein summation convention]] implied). So the manifold ''T''*''M'' itself carries local coordinates (''x''<sup>''i''</sup>, ''p''<sub>''i''</sub>) where the ''x''<nowiki/>'s are coordinates on the base and the ''p's'' are coordinates in the fibre. The canonical one-form is given in these coordinates by :<math>\theta_{(x,p)}=\sum_{i=1}^n p_i \, dx^i.</math> Intrinsically, the value of the canonical one-form in each fixed point of ''T*M'' is given as a [[pullback (differential geometry)|pullback]]. Specifically, suppose that {{nowrap|π : ''T*M'' → ''M''}} is the [[Projection (mathematics)|projection]] of the bundle. Taking a point in ''T''<sub>''x''</sub>*''M'' is the same as choosing of a point ''x'' in ''M'' and a one-form ω at ''x'', and the tautological one-form θ assigns to the point (''x'', ω) the value :<math>\theta_{(x,\omega)}=\pi^*\omega.</math> That is, for a vector ''v'' in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ to ''v'' at (''x'', ω) is computed by projecting ''v'' into the tangent bundle at ''x'' using {{nowrap|''d''π : ''T''(''T''*''M'') → ''TM''}} and applying ω to this projection. Note that the tautological one-form is not a pullback of a one-form on the base ''M''.
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