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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?

Thanks!

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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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    Ah, right. Thanks for pointing that out.2012-12-30