cotangent bundle

Overview

Let $M$ be a differentiable manifold. Analogously to the construction of the tangent bundle, we can make the set of covectors on a given manifold into a vector bundle 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 (http://planetmath.org/NotesOnTheClassicalDefinitionOfAManifold). Let $M$ be an $n$-dimensional differentiable manifold, let $\{V_{\alpha }\mid \alpha \in 𝒜\}$ (each $V_{\alpha }$ is an open subset of ${ℝ}^n$) be an atlas of $M$ with transition functions ${\sigma }_{\alpha \beta }$.

As an atlas for $T^{*}(M)$, we may take $\{V_{\alpha }\times {ℝ}^n\mid \alpha \in 𝒜\}$. We may construct transition functions $\sigma ^{\prime }{}_{\alpha \beta }$ as follows:

$${\left(\sigma ^{\prime }{}_{\alpha \beta }(x^1,\mathrm{\dots },x^{2n})\right)}^i={\left({\sigma }_{\alpha \beta }(x^1,\mathrm{\dots },x^n)\right)}^i\mathit{\hspace{1em}\hspace{1em}}1\le i\le n$$

$${\left(\sigma ^{\prime }{}_{\alpha \beta }(x^1,\mathrm{\dots },x^{2n})\right)}^{i+n}=\sum _{j=1}^n\frac{\partial {\left({\sigma }_{\alpha \beta }(x^1,\mathrm{\dots },x^n)\right)}^i}{\partial x^j}{x}^{j+n}\mathit{\hspace{1em}\hspace{1em}}1\le i\le n$$

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 Structure

The cotangent bundle 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 $\pi$:

$$\pi {(x^1,\mathrm{\dots },x^{2n})}^i=x^i$$

The local trivializations are also somewhat trivial:

$${\varphi }_{\alpha }(x^1,\mathrm{\dots },x^{2n})=x^{i+n}$$

Finally, the transition functions are given as follows:

$$g_{\alpha \beta }{(x^1,\mathrm{\dots },x^{2n})}_j^i=\frac{\partial {\left({\sigma }_{\alpha \beta }(x^1,\mathrm{\dots }{x}^n)\right)}^i}{\partial x^j}$$

For a verification that $(T^{*}M,\pi ,{\varphi }_{\alpha },g_{\alpha \beta })$ satisfies the three criteria for a bundle, please see the attachment.

Properties

The cotangent bundle $T^{*}M$ is the vector bundle dual to the tangent bundle $TM$. On any differentiable manifold, $T^{*}M\cong TM$ (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