cotangent bundle


Let M be a differentiable manifold. Analogously to the construction of the tangent bundleMathworldPlanetmath, we can make the set of covectors on a given manifold into a vector bundleMathworldPlanetmath over M, denoted T*M and called the cotangent bundle of M.

Rigorous Definition

To make this definition precise it is convenient to use the classical definition of a manifold ( Let M be an n-dimensional differentiable manifold, let {Vαα𝒜} (each Vα is an open subset of n) be an atlas of M with transition functionsMathworldPlanetmath σαβ.

As an atlas for T*(M), we may take {Vα×nα𝒜}. We may construct transition functions σαβ as follows:

(σαβ(x1,,x2n))i=(σαβ(x1,,xn))i  1in
(σαβ(x1,,x2n))i+n=j=1n(σαβ(x1,,xn))ixjxj+n  1in

For these to be valid transition functions, they must satisfy the three criteria. For a verification that these criteria are satisfied, please see the attachment.

Bundle StructureMathworldPlanetmath

The cotangent bundleMathworldPlanetmath is a GL(n) vector bundle over the manifold M. To substantiate this claim, we must specify a projection map onto the manifold M and local trivializations and transition functions and verify that they satisfies the defining properties of a bundle. In terms of the local coordinates used above, it is easy to describe the projection map π:


The local trivializations are also somewhat trivial:


Finally, the transition functions are given as follows:


For a verification that (T*M,π,ϕα,gαβ) satisfies the three criteria for a bundle, please see the attachment.


The cotangent bundle T*M is the vector bundle dual to the tangent bundle TM. On any differentiable manifold, T*MTM (for example, by the existence of a Riemannian metric), but this identification is by no means canonical, and thus it is useful to distinguish between these two objects.

The cotangent bundle to any manifold has a natural symplectic structure given in terms of the Poincaré 1-form, which is in some sense unique. This is not true of the tangent bundle. The existence of a symplectic structure implies that the cotangent bundle is always orientable, even if the original manifold is not.

Title cotangent bundle
Canonical name CotangentBundle
Date of creation 2013-03-22 13:59:02
Last modified on 2013-03-22 13:59:02
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 17
Author rspuzio (6075)
Entry type Definition
Classification msc 58A32