Inmathematics, especiallydifferential geometry, thecotangent bundle of asmooth manifold is thevector bundle of all thecotangent spaces at every point in the manifold. It may be described also as thedual bundle to thetangent bundle. This may be generalized tocategories with more structure than smooth manifolds, such ascomplex manifolds, or (in the form of cotangent sheaf)algebraic varieties orschemes. 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 adiagonal mapping${\displaystyle \mathrm{\Delta }}$ andgerms.

Let${\displaystyle M}$ be asmooth manifold and let${\displaystyle M\times M}$ be theCartesian product of${\displaystyle M}$ with itself. Thediagonal mapping${\displaystyle \mathrm{\Delta }}$ sends a point${\displaystyle p}$ in${\displaystyle M}$ to the point${\displaystyle (p,p)}$ of${\displaystyle M\times M}$. The image of${\displaystyle \mathrm{\Delta }}$ is called the diagonal. Let${\displaystyle \mathcal{I}}$ be thesheaf ofgerms of smooth functions on${\displaystyle M\times M}$ which vanish on the diagonal. Then thequotient sheaf${\displaystyle \mathcal{I}/{\mathcal{I}}^2}$ consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. Thecotangent sheaf is defined as thepullback of this sheaf to${\displaystyle M}$:

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

ByTaylor's theorem, this is alocally free sheaf of modules with respect to the sheaf of germs of smooth functions of${\displaystyle M}$. Thus it defines avector bundle on${\displaystyle M}$: thecotangent bundle.

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

A smooth morphism${\displaystyle \varphi :M\to N}$ of manifolds induces apullback sheaf${\displaystyle {\varphi }^{\ast }{T}^{\ast }N}$ onM. There is aninduced map of vector bundles${\displaystyle {\varphi }^{\ast }(T^{\ast }N)\to T^{\ast }M}$.

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${\displaystyle T^{\ast }{\mathbb{R}}^n={\mathbb{R}}^n\times ({\mathbb{R}}^n{)}^{\ast }}$, where${\displaystyle ({\mathbb{R}}^n{)}^{\ast }}$ denotes thedual space of covectors, linear functions${\displaystyle v^{\ast }:{\mathbb{R}}^n\to \mathbb{R}}$.

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

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

where${\displaystyle d{f}_x\in T_x^{\ast }M}$ is thedirectional derivative${\displaystyle d{f}_x(v)=\mathrm{\nabla }f(x)\cdot v}$. By definition, the cotangent bundle in this case is

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

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

${\displaystyle T^{\ast }M=\{(x,v^{\ast })\in T^{\ast }{\mathbb{R}}^n:f(x)=0,v\in T_x{\mathbb{R}}^n,d{f}_x(v)=0\}.}$

Since the cotangent bundle${\displaystyle X=T^{\ast }M}$ is avector bundle, it can be regarded as a manifold in its own right. Because at each point the tangent directions of${\displaystyle M}$ can be paired with their dual covectors in the fiber,${\displaystyle X}$ possesses a canonical one-form${\displaystyle \theta }$ called thetautological one-form, discussed below. Theexterior derivative of${\displaystyle \theta }$ is asymplectic 2-form, out of which a non-degeneratevolume form can be built for${\displaystyle X}$. For example, as a result${\displaystyle X}$ is always anorientable manifold (the tangent bundle${\displaystyle TX}$ is an orientable vector bundle). A special set ofcoordinates can be defined on the cotangent bundle; these are called thecanonical coordinates. Because cotangent bundles can be thought of assymplectic manifolds, any real function on the cotangent bundle can be interpreted to be aHamiltonian; thus the cotangent bundle can be understood to be aphase space on whichHamiltonian mechanics plays out.

The cotangent bundle carries a canonical one-form θ also known as thesymplectic potential,Poincaré1-form, orLiouville1-form. This means that if we regardT*M as a manifold in its own right, there is a canonicalsection of the vector bundleT*(T*M) overT*M.

This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose thatxⁱ are local coordinates on the base manifoldM. In terms of these base coordinates, there are fibre coordinatesp_i : a one-form at a particular point ofT*M has the formp_i dxⁱ (Einstein summation convention implied). So the manifoldT*M itself carries local coordinates (xⁱ,p_i) where thex's are coordinates on the base and thep's are coordinates in the fibre. The canonical one-form is given in these coordinates by

${\displaystyle {\theta }_{(x,p)}=\sum _{i=1}^n{p}_{i}d{x}^i.}$

Intrinsically, the value of the canonical one-form in each fixed point ofT*M is given as apullback. Specifically, suppose thatπ :T*M →M is theprojection of the bundle. Taking a point inT_x*M is the same as choosing of a pointx inM and a one-form ω atx, and the tautological one-form θ assigns to the point (x, ω) the value

${\displaystyle {\theta }_{(x,\omega )}={\pi }^{\ast }\omega .}$

That is, for a vectorv in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ tov at (x, ω) is computed by projectingv into the tangent bundle atx usingdπ :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 baseM.

The cotangent bundle has a canonicalsymplectic 2-form on it, as anexterior derivative of thetautological one-form, thesymplectic potential. Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on${\displaystyle {\mathbb{R}}^n\times {\mathbb{R}}^n}$. But there the one form defined is the sum of${\displaystyle y_{i}d{x}_i}$, and the differential is the canonical symplectic form, the sum of${\displaystyle d{y}_i\wedge d{x}_i}$.

If the manifold${\displaystyle M}$ represents the set of possible positions in adynamical system, then the cotangent bundle${\displaystyle T^{\ast }M}$ can be thought of as the set of possiblepositions andmomenta. For example, this is a way to describe thephase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle. The above symplectic construction, along with an appropriateenergy function, gives a complete determination of the physics of system. SeeHamiltonian mechanics and the article ongeodesic flow for an explicit construction of the Hamiltonian equations of motion.

• 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.

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• Differential topology
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