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In John Lee's Intro to Smooth Manifold book (2003 Springer) , I need some help with an example of an immersion.

On page 156 Example 7.1 c), If $\gamma(t): J \to M$ is a smooth curve ...then $\gamma$ is an immersion if and only if $\gamma'(t)\neq 0 $ for all $t \in J$.

Can someone provide a general explanation for what this $\gamma'(t)\neq 0 $ means? The curve does not "stop"?

How is the $\gamma'(t)\neq 0 $ condition related to the fact that a map $F$ is an immersion if its pushforward $F_*$ is injective? Is $\gamma'(t)$ suppose to be injective (one to one)? Would this mean $\gamma'(t)$ have to be different for each $t$ ? Seems like if $\gamma'(t)=a$ a constant, then it will not be one to one?

In example 7.3 on p157, how is it that:

$$| e^{-2\pi i c n_2}(e^{2\pi ic n_1}-e^{2\pi ic n_2})|=|e^{2\pi ic n_1}-e^{2\pi ic n_2}| $$

I can't figure out what happened to $e^{-2\pi i c n_2}$, am I missing something obvious?

xbl
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  • $d\gamma_t$ (a linear map) needs to be injective for each $t$. This is not the same as saying that $\gamma'$ needs to be injective. In fact, that is trivially true (for most curves) because $\gamma'(t)\in T_{\gamma(t)}M$ lives in different tangent spaces for different values of $t$! – wj32 Jun 16 '13 at 01:17

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Let's see, in page 75 you have the formula $$\gamma'(t_0)=\gamma_*\Big(\frac{d}{dt}\Big|_{t_0}\Big)$$ and we have that $\gamma_* : T_{t_0}\mathbb{R} \to T_{\gamma(t_0)}M$ so this linear transformation is injective if and only if $\gamma_*\Big(\frac{d}{dt}\Big|_{t_0}\Big)\neq 0$ for every $t_0$, therefore $\gamma'(t_0) \neq 0$ for every $t_0$.

Note that $|e^{-2\pi icn_1}|=1$ because $e^{-2\pi i k}= \cos(2\pi k)- i\sin(2\pi k)$ and so, $|e^{-2\pi i k}|^2= \cos^2(2\pi k)+\sin^2(2\pi k)=1$ and you have the desired.

math_man
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