Cotangent bundle

In

. 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.

Formal definition via diagonal morphism

There are several equivalent ways to define the cotangent bundle.

.

Let M be a

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

on M: the cotangent bundle.

one-forms

.

Contravariance properties

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 ${\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M}$.

Examples

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 ${\displaystyle u\in T_{x}M,}$

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

The cotangent bundle as phase space

Since the cotangent bundle X = T*M is a

plays out.

The tautological one-form

The cotangent bundle carries a canonical one-form θ also known as the

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 xⁱ 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 dxⁱ (

Einstein summation convention

implied). So the manifold T*M itself carries local coordinates (xⁱ, 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

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

Intrinsically, the value of the canonical one-form in each fixed point of T*M is given as a pullback. Specifically, suppose that π : T*M → M 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 ω at x, and the tautological one-form θ assigns to the point (x, ω) the value

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

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, ω) is computed by projecting v into the tangent bundle at x using dπ : 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 base M.

Symplectic form

The cotangent bundle has a canonical

. 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 ${\displaystyle y_{i}\,dx_{i}}$, and the differential is the canonical symplectic form, the sum of ${\displaystyle dy_{i}\land dx_{i}}$.

Phase space

If the manifold ${\displaystyle M}$ represents the set of possible positions in a dynamical system, then the cotangent bundle ${\displaystyle \!\,T^{*}\!M}$ can be thought of as the set of possible positions and momenta. For example, this is a way to describe the

for an explicit construction of the Hamiltonian equations of motion.

See also

• Legendre transformation

References

• 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.