Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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).

share|cite|improve this question
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
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?)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.