Tangent-bundle cocycle from coordinate changes

Statement

Let $\{(U_i,\varphi_i)\}$ be a smooth atlas on an $n$ -manifold $M$ . On $U_i\cap U_j$ , define the change from $j$ -th coordinate components to $i$ -th coordinate components by

A_{ij}(x) =D(\varphi_i\circ\varphi_j^{-1})_{\varphi_j(x)} \in\operatorname{GL}(n,\mathbb R).

Then

A_{ii}=I, \qquad A_{ji}=A_{ij}^{-1}, \qquad A_{ik}=A_{ij}A_{jk}.

Thus the Jacobians form a smooth $\operatorname{GL}(n,\mathbb R)$ -valued Čech $1$ -cocycle, and the vector bundle obtained by gluing $U_i\times\mathbb R^n$ with this cocycle is the tangent bundle $TM$ .

Why the cocycle law holds

On a triple overlap,

\varphi_i\circ\varphi_k^{-1} =(\varphi_i\circ\varphi_j^{-1}) \circ(\varphi_j\circ\varphi_k^{-1}).

The chain rule therefore gives $A_{ik}=A_{ij}A_{jk}$ . In this example, the bundle cocycle is not extra arbitrary data: it is functorially derived from the smooth atlas.

Frame-bundle interpretation

Coordinate vector fields give local frames of $TM$ . Their transition matrices are the same Jacobian data, subject to the chosen direction convention. The associated frame bundle is a principal $\operatorname{GL}(n,\mathbb R)$ -bundle.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: tangent bundles, coordinate frames, and transformation laws.