Math Help - integral curve

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

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 a,b)->M" alt="ca,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 and are diffeomorphisms. with sufficient small.

So we can define the topology as: is open, if , is open in (a,b).
Then and and the corresponding inverse maps are all continuous.

And the smooth structure can be defined as
if then we can choose the coordinate system with \phi a coordinate
and analog if q=c(p).

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

Regards