# 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 $\{V_{\alpha}\mid\alpha\in{\cal A}\}$ (each $V_{\alpha}$ is an open subset of $\mathbb{R}^{n}$) be an atlas of $M$ with transition functions $\sigma_{\alpha\beta}$.

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

 $\bigg{(}{\sigma^{\prime}}_{\alpha\beta}(x^{1},\ldots,x^{2n})\bigg{)}^{i}=\bigg% {(}\sigma_{\alpha\beta}(x^{1},\ldots,x^{n})\bigg{)}^{i}\qquad 1\leq i\leq n$
 $\bigg{(}{\sigma^{\prime}}_{\alpha\beta}(x^{1},\ldots,x^{2n})\bigg{)}^{i+n}=% \sum_{j=1}^{n}{\partial\bigg{(}\sigma_{\alpha\beta}(x^{1},\ldots,x^{n})\bigg{)% }^{i}\over\partial x^{j}}x^{j+n}\qquad 1\leq i\leq n$

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 $\pi$:

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

The local trivializations are also somewhat trivial:

 ${\phi_{\alpha}(x^{1},\ldots,x^{2n})}=x^{i+n}$

Finally, the transition functions are given as follows:

 $g_{\alpha\beta}(x^{1},\ldots,x^{2n})^{i}_{j}={\partial\big{(}\sigma_{\alpha% \beta}(x^{1},\ldots x^{n})\big{)}^{i}\over\partial x^{j}}$

For a verification that $(T^{*}M,\pi,\phi_{\alpha},g_{\alpha\beta})$ 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 CotangentBundle 2013-03-22 13:59:02 2013-03-22 13:59:02 rspuzio (6075) rspuzio (6075) 17 rspuzio (6075) Definition msc 58A32