In this entry, we shall verify that the transition functions proposed for the cotangent bundle the three criteria required by the classical definition of a manifold.

The first criterion is the easiest to verify. If $\alpha = \beta$ , then $\sigma_{\alpha \alpha}$ reduces to the identity and we have $$\bigg({\sigma'}_{\alpha \alpha} (x_1, \ldots, x_{2n}) \bigg)^i = \bigg(\sigma_{\alpha \alpha} (x_1, \ldots, x_n) \bigg)^i = x^i \qquad 1 \le i \le n $$ $$\bigg({\sigma'}_{\alpha \alpha} (x_1, \ldots, x_{2n}) \bigg)^{i+n} = \sum_{j = 1}^n {\partial \bigg(\sigma_{\alpha \alpha} (x_1, \ldots, x_n) \bigg)^i \over \partial x_j} x^{j+n} = \sum_{j = 1}^n {\partial x^i \over \partial x_j} x^{j+n} = x^{i+n} \qquad 1 \le i \le n$$ Thus we see that ${\sigma'}_{\alpha \alpha}$ is the identity map, as required.

Next, we turn our attention to the third criterion -- showing that ${\sigma'}_{\beta \gamma} \circ {\sigma'}_{\alpha \beta} = {\sigma'}_{\alpha \gamma}$ . For clarity of notation let us define $y^i = ({\sigma'}_{\alpha \beta})^i (x^1, \ldots x^{2n})$ . Then we have \begin{eqnarray*} ({\sigma'}_{\beta \gamma} \circ {\sigma'}_{\alpha \beta})^i (x^1, \dots, x^{2n}) &=& ({\sigma'}_{\beta \gamma})^i (y^1, \dots, y^{2n}) \\ &=& (\sigma_{\beta \gamma})^i (y^1, \dots, y^n) \\ &=& (\sigma_{\beta \gamma} \circ \sigma_{\alpha \beta})^i (x^1, \dots, x^n) \\ &=& (\sigma_{\alpha \gamma})^i (x^1, \dots, x^n) \\ &=& ({\sigma'}_{\alpha \gamma})^i (x^1, \dots, x^{2n}) \\ \end{eqnarray*}when $1 \le i \le n $ . \begin{eqnarray*} ({\sigma'}_{\beta \gamma} \circ {\sigma'}_{\alpha \beta})^{i+n} (x^1, \dots, x^{2n}) &=& ({\sigma'}_{\beta \gamma})^{i+n} (y^1, \dots, y^{2n}) \\ &=& \sum_{j = 1}^n {\partial \bigg(\sigma_{\beta \gamma} (y_1, \ldots, y_n) \bigg)^i \over \partial y_j} y^{j+n} \\ &=& \sum_{j = 1}^n \sum_{k = 1}^n {\partial \bigg(\sigma_{\beta \gamma} (y_1, \ldots, y_n) \bigg)^i \over \partial y_j} {\partial \bigg(\sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^j \over \partial x_k} x^{n+k} \\ &=& \sum_{k = 1}^n {\partial \bigg(\sigma_{\beta \gamma} \circ \sigma_{\alpha \beta} (x_1, \ldots, x_n) \bigg)^i \over \partial x_k} x^{n+k} \\ &=& \sum_{k = 1}^n {\partial \bigg(\sigma_{\alpha \gamma} (x_1, \ldots, x_n) \bigg)^i \over \partial x_k} x^{n+k} \\ &=& {\sigma'}_{\alpha \gamma} (x^1, \dots, x^{2n}) \\ \end{eqnarray*}when $1 \le i \le n $ .

Finally, the second criterion does not need to be checked because it is a consequence of the first and third criteria.