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# 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. 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.

## Mathematics Subject Classification

58A32*no label found*

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