Editing Cotangent bundle (section)

== Formal definition via [[diagonal morphism]] ==
There are several equivalent ways to define the cotangent bundle. [[Cotangent sheaf#Construction through a diagonal morphism|One way]] is through a [[diagonal mapping]] &Delta; and  [[germ (mathematics)|germs]].

Let ''M'' be a [[Differentiable manifold|smooth manifold]] and let ''M''&times;''M'' be the [[Cartesian product]] of ''M'' with itself.  The [[diagonal mapping]] &Delta; sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''&times;''M''.  The image of &Delta; is called the diagonal.  Let <math>\mathcal{I}</math> be the [[sheaf (mathematics)|sheaf]] of [[germ (mathematics)|germs]] of smooth functions on ''M''&times;''M'' which vanish on the diagonal.  Then the [[sheaf (mathematics)#Operations|quotient sheaf]] <math>\mathcal{I}/\mathcal{I}^2</math> consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.  The [[cotangent sheaf]] is defined as the [[inverse image functor|pullback]] of this sheaf to ''M'':

:<math>\Gamma T^*M=\Delta^*\left(\mathcal{I}/\mathcal{I}^2\right).</math>

By [[Taylor's theorem]], this is a [[locally free sheaf]] of modules with respect to the sheaf of germs of smooth functions of ''M''.  Thus it defines a [[vector bundle]] on ''M'':  the '''cotangent bundle'''.

[[Smooth function|Smooth]] [[Section (fiber bundle)|sections]] of the cotangent bundle are called (differential) [[one-form]]s.