integral curve

**Sogan** · Jan 10th 2011, 10:51 AM

Hello,

Let M be a manifold
I want to show, that the integral curves of a vector field V are immersed submanifolds of M.

We assume that the vector field V doesn't vanish anywhere.
I could show that any integral curve is a immersion.

If the integral curve c is also injective, that is c is an injective immersion then the image of c is a immersed submanifold.

But what happens if c isn't injective. For instance if c is periodic?
How can i show that the image of c is a immersed submanifold, i.e. the inclusion map i: c(a,b)->M is a smooth immersion?

Please help me!

Thanks in advance

**Sogan** · Jan 10th 2011, 01:11 PM

i have a new idea how to show that a periodic integral curve is a immersed submanifold, but i need a little help.

My Idea is the following:

if $c<img src=$ a,b)->M" alt="c

a,b)->M" /> is a periodic integral curve with period p.

then i need to define a topology and a smooth structure on the image c(a,b), s.t. c(a,b) is a manifold and the inclusion map i:c(a,b)->M is a smooth immersion.

I think i need to define the topology, s.t. the maps $c_{|(0,p)}$ and $c_{|(\epsilon, \epsilon +p)}$ are diffeomorphisms. with $\epsilon >0$ sufficient small.

So we can define the topology as: $U\subset c(a,b)$ is open, if $c^{-1}(U \cap c(0,p))$ , $c^{-1}(U \cap c(\epsilon,\epsilon + p))$ is open in (a,b).
Then $c_{|(0,p)}$ and $c_{|(\epsilon, \epsilon +p)}$ and the corresponding inverse maps are all continuous.

And the smooth structure can be defined as
if $q \in c_{|(0,p)} \subset c(a,b)$ then we can choose the coordinate system $\phi \circ c_{|(0,p)}^{-1}$ with \phi a coordinate
and analog if q=c(p).

Do you mean, that this is enough or right??

Regards

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