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).
Is injection from manifold to tangent manifold well defined?
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differential-geometry
manifolds
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2Yes, this is the [zero section](http://en.wikipedia.org/wiki/Vector_bundle#Sections_and_locally_free_sheaves), which exists in every vector bundle. – 2012-04-23
1 Answers
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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?)