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Version 6 |
| {\bf Overview} |
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| 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 $M$. |
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 $M$. |
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| {\bf Rigorous Definition} |
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| 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 $\sigma_{\alpha \beta}$. |
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 $\sigma_{\alpha \beta}$. |
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| 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: |
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: |
| $$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_n, y_1, \ldots, y_n) \bigg)^i = \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \qquad 1 \le i \le n$$ |
$$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_n, y_1, \ldots, y_n) \bigg)^i = \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \qquad 1 \le i \le n$$ |
| $$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_n, y_1, \ldots, y_n) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \over \partial x_j} y^j \qquad 1 \le i \le n$$ |
$$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_n, y_1, \ldots, y_n) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \over \partial x_j} y^j \qquad 1 \le i \le n$$ |
| 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. |
For thehe to be valid transition functions, they must satisfy the three criteria. For a verification that these criteria are satisfied, please see the attachment. |
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| {\bf Properties} |
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| 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 $T^*M$ 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. |
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| The cotangent bundle to any manifold has a natural symplectic structure, 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. |
{\bf Symplectic structure} |
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For example, the cotangent bundle to any manifold has a natural symplectic structure, 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. |
