Periodic curve as immersed submanifold

[robforsub]

Is a periodic curve still an immersed submanifold of a manifold M? Suppose y is the curve
map an interval to a manifold M, and y is periodic, which means it is not injective. And immersed submanifold must be the image of a injective immersion.

 

[Tinyboss]

If it's periodic, maybe you want to immerse S^1 instead of an interval.

 

[robforsub]

Let me clarify it a little bit, so I want to show that the image of that curve y is a immersed submanifold of manifold M, and without the condition y'(t)!=0 for all t, I can not say y is a immersion, then the image of curve y is immersed submanifold, right?

 

[Tinyboss]

What I mean is that, if y' is everywhere nonzero, then y is an immersion. And although y isn't injective, its image may still be an immersed submanifold of M, since a periodic map from R to M descends to a map Y from S^1 to M, so that going once around the circle covers a single period. And if y is injective during each period, then Y is an injective immersion.