# 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 $\{{V}_{\alpha}\mid \alpha \in \mathcal{A}\}$ (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 $\{{V}_{\alpha}\times {\mathbb{R}}^{n}\mid \alpha \in \mathcal{A}\}$. 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 |