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This question was asked in my quiz on smooth manifolds and I was unable to solve it. I tried it again at home and still not able to solve it. Question is:

Let $\sigma$ be an integral curve for a vector field X on a manifold M, with $\sigma(0)=p$. If $X_p\neq 0$, show that there exists $\epsilon >0$ such that $\sigma : (-\epsilon , \epsilon) \to M $ is an immersion.

What I could prove is that: If $p\in M$ then there exists $\epsilon >0$ and an integral curve of X such that $\sigma : (-\epsilon ,\epsilon)\to M$ with $\sigma(0)=p$.

But How can I prove that a $\sigma$ also exists which is an immersion.

Can you please tell?

1 Answers1

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First of all one has to be clear what does it means integral curve, so it would be useful to remember this concept:

An integral curve of starting point $p\in M$ of a vector field $X\colon M\to TM$ is a curve $ \sigma \colon (-\epsilon, \epsilon)\to M$ such that

$\sigma(0)=p$

$\sigma’(t)=X(\sigma(t))$

The first important theorem is the following one:

There exists always a unique integral curve of starting point $p$ related to a vector field $X$

The proof of the theorem is quite easy, because in a coordinate chart $(U,\phi)$, the condition became

$(\sigma\circ \phi)^{-1}(0)=0$

$(\sigma\circ\phi^{-1})’(t)= \sum_i d\phi_{i,\sigma(t)}(X(\sigma(t))\frac{\partial}{\partial \phi_{i,\sigma(t)}}$

In other words

$(\sigma\circ \phi)^{-1}(0)=0$

$\frac{d}{dt}(\sigma\circ\phi^{-1})_i= d\phi_{i,\sigma(t)}(X(\sigma(t))$ for each $i=1,\dots n$

This is a Cauchy problem in $t,y_1=\sigma\circ\phi^{-1}_1,\cdots ,y_n= \sigma\circ \phi^{-1}_n$ variables and so by locally existence and uniqueness Cauchy-Lipschitz theorem (or equivalently Picard-Lindelöf theorem) there exists a unique solution $\gamma(t)\colon (-\epsilon, \epsilon)\to \phi(U)$ for an $\epsilon$ enough small.

Then our integral curve will be $\sigma:=\phi^{-1} \circ \gamma $ and we have done.

But now you have an important condition to your problem:

$X_p\neq 0$

So by continuity (X is a smooth map) of $X$ there exists an appropriate neighbourhood of $p$ such that $X_q\neq 0$. If you intersect your integral curve $\sigma$ with that neighbourhood then you will get $d\sigma_t(d/dt)=\sigma’(t)=X(\sigma(t))\neq 0$

But $d/dt$ is a basis of $T_t(-\epsilon,\epsilon)$ and so the differential of $\sigma$ in $t$, $d\sigma_t$, has to be injective, that is the definition of a smooth map to be an immersion.

Thus we have done. If there is some mistake or error please tell me and we will discuss

Federico Fallucca
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  • Man, thank you very much for answering. I have asked many other questions on smooth manifolds on this site and unfortunately people are not answering. Can you please answer some of them in your spare time. I live in a very poor country where I don't have any guidence. –  Jul 04 '22 at 09:25
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    @Avenger i can try, can you send me the link of one question here? – Federico Fallucca Jul 04 '22 at 09:34
  • Thank you very much. You are very kind. I am sending the link of 1 question here. –  Jul 04 '22 at 09:35
  • https://math.stackexchange.com/questions/4443064/smooth-functions-and-pullback-map It has a bounty of 50 points. –  Jul 04 '22 at 09:35
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    @Avenger i tried to answer, I hope will be clear. Few suggestions: (1) Is better to ask just one question per time. I need to take 3 hours to write the answer, so it makes sense that the people will not answer. (2) specify the definition that you know of the stuff that you are asking for. In that way will be more familiar to answer for the other people. Bye bye – Federico Fallucca Jul 04 '22 at 12:42
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    Thank you very much mate for your help! –  Jul 04 '22 at 12:59
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    @Avenger if you can and want, can you verify my answer here? :) – Federico Fallucca Jul 04 '22 at 13:02
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    Will surely soon, Last few months were the worst in my life till now. So, I haven't had a look at many answers. I am going through a lot. –  Jul 04 '22 at 13:04
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    @Avenger ah I see, I’m sorry. Take your time – Federico Fallucca Jul 04 '22 at 13:40
  • @Avenger if you have some other interesting questions, tell me – Federico Fallucca Jul 05 '22 at 11:57
  • Thank you very much! You can have a look at these:https://math.stackexchange.com/questions/4444264/1-form-on-compact-manifold –  Jul 05 '22 at 12:11
  • and this:https://math.stackexchange.com/questions/4443610/every-w-in-omega2-v-is-decomposable-if-operatornamedimv-3 –  Jul 05 '22 at 12:12
  • and this:https://math.stackexchange.com/questions/4439265/any-2-tensor-sum-of-a-symmetric-2-tensor-an-alternating-2-tensor –  Jul 05 '22 at 12:12
  • Thank you very much for the help provided! –  Jul 05 '22 at 12:13
  • If you have some spare time in coming days you can have a look at this:https://math.stackexchange.com/questions/4432668/mathbbc-spectra-homeomorphism-immersion-and-fibre –  Jul 06 '22 at 07:39
  • Hi! I am sorry for replying so late, my physical health has been very precarious, how does $d/dt$ being a basis of $T_t(-\epsilon,, \epsilon)$ implies that differential of $\sigma$ in t will be injective?( $3rd line from below of the answer). –  Dec 02 '22 at 15:55