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I would like to know if the injection map $i : M\to TM$, given by $i(x)\mapsto (x,0)$, is a well-defined and canonical application (not dependent on any particular coordinate chart).

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Yes, this is the zero section, which exists in every vector bundle. –  t.b. Apr 23 '12 at 14:14
    
@t.b. Thank you –  jet Apr 23 '12 at 14:22

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up vote 2 down vote accepted

The answer is "yes." You should prove it yourself as an exercise.

(Hint: the transition functions are smooth maps $\theta_{UV}:U\cap V\to Gl(n;\mathbb{R})$. In particular, for all $x\in U\cap V$, $\theta_{UV}(x)$ is a linear map. What does this say about how $i$ transforms between coordinate systems?)

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