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=                          Cotangent bundle                          =
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                            Introduction
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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.


                        The cotangent sheaf
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Smooth sections of the cotangent bundle are differential one-forms.


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

:\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.

See also: bundle of principal parts (which generalizes the above
constructions to higher orders.)


Contravariance in manifolds
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A smooth morphism  \phi\colon M\to N of manifolds, induces a pullback
sheaf \phi^*T^*N on 'M'. There is an induced map of vector bundles
\phi^*(T^*N)\to T^*M.


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

Given a smooth manifold M\subset \mathbb{R}^n embedded as the
vanishing locus of a smooth function f, its tangent bundle is:

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

where  df_x is the covector defined by the directional derivative
df_x(v)=\tfrac{\partial f}{\partial v}(x)=\nabla\! f(x)\cdot v. Its
cotangent bundle consists of pairs (x,v^*\,\text{mod}\
\mathbb{R}\,df_x), where f(x)=0 and we take the covector v^* in the
quotient space of (\mathbb{R}^n)^* modulo the line generated by df_x.
Of course, the dot product identifies the quotient space
(\mathbb{R}^n)^*/\mathbb{R}\,df_x with the orthogonal space to the
gradient \nabla f(x), so the two bundles are isomorphic.

For example, let 'M' be the 3-sphere given by the vanishing locus of
x^2 + y^2 + z^2 + w^2 = 1 in \mathbb{R}^4. Its tangent bundle is the
set of (x,y,z,w\,;q,r,s,t) with x^2 + y^2 + z^2 + w^2 = 1 and
df_x(q,r,s,t)=2xq+2yr+2zs+2wt=0. Its cotangent bundle is the set of
(x,y,z,w\,; q\,dx+r\,dy+s\,dz+t\,dw) with the covector considered
modulo the direction df_x = 2x\,dx+2y\,dy+2z\,dz+2w\,dw; so at
(x,y,z,w)=(0,\tfrac 35,\tfrac 45,1), we take q\,dx+r\,dy+s\,dz+t\,dw
modulo \mathbb{R}(\tfrac 65dy+\tfrac 85dz+2\,dw ).


                The cotangent bundle as phase space
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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.


The tautological one-form
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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 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''i' are local
coordinates on the base manifold 'M'.  In terms of these base
coordinates, there are fibre coordinates 'p''i':  a one-form at a
particular point of 'T'*'M' has the form 'p''i' 'dx''i' (Einstein
summation convention implied).  So the manifold 'T'*'M' itself carries
local coordinates ('x''i', 'p''i') where the 'x''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

:\theta_{(x,p)}=\sum_{{\mathfrak i}=1}^n p_i \, dx^i.

Intrinsically, the value of the canonical one-form in each fixed point
of 'T*M' is given as a pullback.  Specifically, suppose that  is the
projection of the bundle. Taking a point in 'T''x'*'M' is the same as
choosing of a point 'x' in 'M' and a one-form � at 'x', and the
tautological one-form θ assigns to the point ('x', �) the value

:\theta_{(x,\omega)}=\pi^*\omega.

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',
�) is computed by projecting 'v' into the tangent bundle at 'x' using
and applying � to this projection.  Note that the tautological
one-form is not a pullback of a one-form on the base 'M'.


Symplectic form
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The cotangent bundle has a canonical symplectic 2-form on it, as an
exterior derivative of the tautological one-form, the symplectic
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 \mathbb{R}^n \times \mathbb{R}^n. But there the one form
defined is the sum of y_i\,dx_i, and the differential is the canonical
symplectic form, the sum of dy_i \land dx_i.


Phase space
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If the manifold M represents the set of possible positions in a
dynamical system, then the cotangent bundle \!\,T^{*}\!M can be
thought of as the set of possible 'positions' and 'momenta'. For
example, this is a way to describe the phase 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
appropriate energy function, gives a complete determination of the
physics of system. See Hamiltonian mechanics and the article on
geodesic flow for an explicit construction of the Hamiltonian
equations of motion.


                              See also
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* Legendre transformation


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