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I want to prove that the tangent bundel TM of a manifold is also manifold of dimension twice the dimension of M? could you help me?


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Suppose that $M$ is an $n$-dimensional manifold, with atlas consisting of charts $\{U_\alpha, \phi_\alpha\}$. Let $\psi: \pi^{-1}(U_\alpha) \to U_{\alpha} \times \mathbb{R}^n$ be defined by

$$\psi : [\gamma] \to (\gamma(0), (\psi_{\alpha} \circ \gamma )' (0))$$

Then define a chart on $TM$ by

$$\pi^{-1}(U) \to^{\psi} U \times \mathbb{R}^n \to^{\phi_{\alpha} \times id} \mathbb{R}^{2n}$$

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It's not really true that $U_\alpha \times \Bbb R^n \subset TM$. Instead, there is a possibly smaller open set $V_\alpha \subset U_\alpha$ such that $\pi^{-1}(V_\alpha) \cong V_\alpha \times \Bbb R^n$. So you need to consider the bundle charts here. – Henry T. Horton Dec 30 '12 at 20:16
Ah, right. Thanks for pointing that out. – Isaac Solomon Dec 30 '12 at 22:00

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