If M, N are smooth manifolds and [imath]f: M \rightarrow N[/imath] is [imath]C^{\infty}[/imath], is [imath]f(M)[/imath] a submanifold of N?

[MathNugget]

I am using the definition of embedded submanifolds, and this definition of embedding (a smooth injective map with its derivative injective everywhere).

I am fairly certain f being [imath]C^{\infty}[/imath] isn't enough for the properties above to be fulfilled. How would I produce a counterexample? Would [imath]f:\mathbb{R} \rightarrow \mathbb{R}, f(x)=x^2[/imath] work? I think it is [imath]C^{\infty}[/imath], while its map isn't injective, nor is the derivative injective in 0...

 

[blamocur]

Derivative of [imath]x^2[/imath] is not injective. Also, in the general case, what if the image of [imath]f[/imath] self-intersects? E.g [imath]f:\mathbb R\rightarrow \mathbb R^2[/imath] where [imath]f(t) = (t^3-t, t^2)[/imath].

 

[MathNugget]

Derivative of [imath]x^2[/imath] is not injective.

Yes. I am trying to produce a counterexample, to prove the answer to the question in title is No.

Also, in the general case, what if the image of [imath]f[/imath] self-intersects? E.g [imath]f:\mathbb R\rightarrow \mathbb R^2[/imath] where [imath]f(t) = (t^3-t, t^2)[/imath].

Sounds about right, but then I'd have to prove that function self intersects, I suppose in (0, 0).

 

[blamocur]

Yes. I am trying to produce a counterexample, to prove the answer to the question in title is No.

But since the derivative of [imath]x^2[/imath] is not injective (at 0) it is not really a counterexample.

Sounds about right, but then I'd have to prove that function self intersects, I suppose in (0, 0).

At [imath]t=1[/imath] and [imath](x,y) = (0,1)[/imath]:

 

[MathNugget]

But since the derivative of [imath]x^2[/imath] is not injective (at 0) it is not really a counterexample.

But...I think it is still [imath]C^{\infty}[/imath]. Maybe I am misunderstanding it

I agree your example is correct, and it answers my question... but I'd like to understand why the other one isn't a good counterexample. Also thank you so much for the help.

 

[blamocur]

But...I think it is still [imath]C^{\infty}[/imath]. Maybe I am misunderstanding it

I agree your example is correct, and it answers my question... but I'd like to understand why the other one isn't a good counterexample. Also thank you so much for the help.

This would depend on the specific definition of submanifolds. The links you provided have multiple definitions -- which one are you using?

 

[MathNugget]

This would depend on the specific definition of submanifolds. The links you provided have multiple definitions -- which one are you using?

Well, isn't

this definition of embedding (a smooth injective map with its derivative injective everywhere).

one of the definitions of a submanifold?

In wikipedia terms, it would be an immersion that is an an embedding in the topological sense. Then, an embedding in the topological sense is described as "an injective, continuous map between 2 topological spaces X, Y, that yields a homeomorphism from X to it's image". (sort of).

 

[blamocur]

Well, isn't

one of the definitions of a submanifold?

In wikipedia terms, it would be an immersion that is an an embedding in the topological sense. Then, an embedding in the topological sense is described as "an injective, continuous map between 2 topological spaces X, Y, that yields a homeomorphism from X to it's image". (sort of).

Which means that self-intersections aren't allowed, and thus the answer to your question is "no".

I didn't realize that your initial question didn't even require an injective derivative, i.e., your [imath]f[/imath] is not even an immersion. Which means that [imath]x^2[/imath] is indeed a good counterexample to your initial statement/question. My example is meant to show how not every immersion is an embedding.

Am I making sense?

 

[MathNugget]

Which means that self-intersections aren't allowed, and thus the answer to your question is "no".

I didn't realize that your initial question didn't even require an injective derivative, i.e., your [imath]f[/imath] is not even an immersion. Which means that [imath]x^2[/imath] is indeed a good counterexample to your initial statement/question. My example is meant to show how not every immersion is an embedding.

Am I making sense?

Thank you. I realize now, much appreciated. I'll note it down, I could see it being on the next exam.