Cotangent bundle

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:

${\displaystyle \Gamma T^{*}M=\Delta ^{*}\left({\mathcal {I}}/{\mathcal {I}}^{2}\right).}$

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

${\displaystyle TM=\{(x,v)\in T\,\mathbb {R} ^{n}\$ :\ f(x)=0,\ \,df_{x}(v)=0\},}

where ${\displaystyle df_{x}\in T_{x}^{*}M}$ is the directional derivative ${\displaystyle df_{x}(v)=\nabla \!f(x)\cdot v}$. By definition, the cotangent bundle in this case is

${\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\$ :\ f(x)=0,\ v^{*}\in T_{x}^{*}M{\bigr \}},}

where ${\displaystyle T_{x}^{*}M=\{v\in T_{x}\mathbb {R} ^{n}\$ :\ df_{x}(v)=0\}^{*}.} Since every covector ${\displaystyle v^{*}\in T_{x}^{*}M}$ corresponds to a unique vector ${\displaystyle v\in T_{x}M}$ for which ${\displaystyle v^{*}(u)=v\cdot u,}$ for an arbitrary ${\displaystyle u\in T_{x}M,}$

${\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\$ :\ f(x)=0,\ v\in T_{x}\mathbb {R} ^{n},\ df_{x}(v)=0{\bigr \}}.}

Since the cotangent bundle X = T*M is a vector bundle, it can be regarded as a manifold in its own right. Because at each point the tangent directions of M can be paired with their dual covectors in the fiber, X possesses a canonical one-form θ called the tautological one-form, discussed below. The exterior derivative of θ is a symplectic 2-form, out of which a non-degenerate volume form can be built for X. For example, as a result X is always an orientable manifold (the tangent bundle TX is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates. Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.

• Legendre transformation

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN 0-8053-0102-X.
• Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-63654-4.
• Singer, Stephanie Frank (2001). Symmetry in Mechanics: A Gentle Modern Introduction. Boston: Birkhäuser.