| Version 9 |
Version 8 |
| {\bf Overview} |
{\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 $M$. |
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} |
{\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 $\sigma_{\alpha \beta}$. |
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: |
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: |
|
$$\bigg({\sigma'}_{\alpha \beta} (x^1, \ldots, x^{2n}) \bigg)^i = \bigg(\sigma_{\alpha \beta} (x^1, \ldots, x^n) \bigg)^i \qquad 1 \le i \le n$$
|
$$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_{2n}) \bigg)^i = \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \qquad 1 \le i \le n$$
|
|
$$\bigg({\sigma'}_{\alpha \beta} (x^1, \ldots, x^{2n}) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \beta} (x^1, \ldots, x^n) \bigg)^i \over \partial x^j} x^{j+n} \qquad 1 \le i \le n$$
|
$$\bigg({\sigma'}_{\alpha \beta} (x_1, \ldots, x_{2n}) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \over \partial x_j} x^{j+n} \qquad 1 \le i \le n$$
|
| 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. |
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: |
|
| $$g_{\alpha \beta} (x^1, \ldots, x^{2n})^i_j = {\partial (\sigma (x^1, \ldots x^n))^i \over \partial x^j}$$ |
|
|
|
|
|
| {\bf Properties} |
{\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 $T^*M$ 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. |
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. |
