In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.

There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs.

Let M be a smooth manifold and let M×M be the Cartesian product of M with itself. The diagonal mapping Δ sends a point p in M to the point (p,p) of M×M. The image of Δ is called the diagonal. Let ${\displaystyle {\mathcal {I}}}$ be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf ${\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}$ consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M:

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 sections of the cotangent bundle are called (differential) one-forms.

A smooth morphism ${\displaystyle \phi \colon M\to N}$ of manifolds induces a pullback sheaf ${\displaystyle \phi ^{*}T^{*}N}$ on M. There is an induced map of vector bundles ${\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M}$.

The tangent bundle of the vector space ${\displaystyle \mathbb {R} ^{n}}$ is ${\displaystyle T\,\mathbb {R} ^{n}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}}$, and the cotangent bundle is ${\displaystyle T^{*}\mathbb {R} ^{n}=\mathbb {R} ^{n}\times (\mathbb {R} ^{n})^{*}}$, where ${\displaystyle (\mathbb {R} ^{n})^{*}}$ denotes the dual space of covectors, linear functions ${\displaystyle v^{*}:\mathbb {R} ^{n}\to \mathbb {R} }$.

Given a smooth manifold ${\displaystyle M\subset \mathbb {R} ^{n}}$ embedded as a hypersurface represented by the vanishing locus of a function ${\displaystyle f\in C^{\infty }(\mathbb {R} ^{n}),}$ with the condition that ${\displaystyle \nabla f\neq 0,}$ the tangent bundle is