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'cotangent bundle'
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| Title of object: |
cotangent bundle |
| Canonical Name: |
CotangentBundle |
| Type: |
Definition |
| Created on: |
2003-10-06 04:58:21 |
| Modified on: |
2004-12-21 16:02:17 |
| Classification: |
msc:58A32 |
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Content:
{\bf 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 {\em cotangent} bundle of $M$.
{\bf Rigorous Definition}
To make this definition precise it is convenient to use the \PMlinkname{classical definition of a manifold}{NotesOnTheClassicalDefinitionOfAManifold}. Let $M$ be an $n$-dimensional differentiable manifold, let $\{V_\alpha \mid \alpha {\cal A}\}$ (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 ${\sigma'}_{\alpha \beta}$ as follows:
$$\bigg({\sigma'}_{\alpha \beta} (x^1, \ldots, x^{2n}) \bigg)^i = \bigg(\sigma_{\alpha \beta} (x^1, \ldots, x^n) \bigg)^i \qquad 1 \le i \le n$$
$$\bigg({\sigma'}_{\alpha \beta} (x^1, \ldots, x^{2n}) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \beta} (x^1, \ldots, x^n) \bigg)^i \over \partial x^j} x^{j+n} \qquad 1 \le i \le n$$
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.
{\bf 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 $\pi$:
$$\pi (x^1, \ldots, x^{2n}) = (x^1, \ldots, x^n)$$
The local trivializations are also somewhat trivial:
$$\phi_\alpha (x^1, \ldots, x^{2n}) = (x^n, \ldots, x^{2n})$$
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 second attachment.
{\bf 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 Poincar\'e 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. |
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