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.

 

[blue_raver22]

Yes, a periodic curve can still be considered an immersed submanifold of a manifold M. The definition of an immersed submanifold is that it is the image of an injective immersion, and while a periodic curve is not injective, it can still be the image of an injective immersion. This is because an immersion maps a manifold to another manifold, and the periodic curve is mapped to the same manifold M. The periodicity of the curve does not affect its status as an immersed submanifold, as long as it is the image of an injective immersion. Therefore, a periodic curve can be considered an immersed submanifold of a manifold M.