M. Spivak, problem 25 chapter 2

[bigli]

How can I show that the set {(x,|x|) , x in real numbers} is not the image of any immersion of R into R^2 ?

problem 25 chapter 2 differential geometry M. Spivak

 

[christoff]

As per the forum rules, nobody should be helping you with this unless you try working the problem first, and showing us where you got stuck. Moreover, it may help to cite the definition you are using for immersion when you begin.

 

[bigli]

http://trainbit.com/files/0810149884/Emb_Submanifold.jpg

 

[HallsofIvy]

Have you actually looked at the image of (x, |x|) in R₂ (that is, the graph of y= |x|). Can you see why it is NOT a smooth manifold? That should tell you what point to focus on in your proof.

 

[bigli]

Please! think about and answer to main problem in my first post and attend to my notes in my second post.

 

[christoff]

Please! think about and answer to main problem in my first post and attend to my notes in my second post.

HallsofIvy has told you exactly how to go about this. If there existed an immersion whose image was the set $\{(x,|x|)\}$, then this set would have a smooth manifold structure. So to prove that no immersion exists, it is sufficient to prove that that you cannot define a smooth manifold structure on $\{(x,|x|)\}$.

I know you said in your "attempt" at a proof that you didn't want to look at the point x=0, but my friend, this will be necessary, since this point is the reason no such immersion exists.