Example of differentiable manifold of class C^1

[mnb96]

Hello,

I read from several sources the statement that the set of points M[itex]\in[/itex]ℝ² given by [itex](t, \, |t|^2)[/itex] is an example of differentiable manifold of class C¹ but not C².

Is that true?
To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should be able to find an atlas of charts [itex]x_i:U_i \rightarrow M[/itex] such that all the points of M are covered by the atlas.
So how do we cover the point (0,0) ?

Do we need to use 3 charts as follows?

[tex]x_1(t) = (t,\, t^2) \quad \, t\in(0,+\infty)[/tex]
[tex]x_2(t) = (t,\, -t^2) \quad \, t\in(-\infty, 0)[/tex]
[tex]x_3(t) = (t,t^2) \quad \, t\in(-1,1)[/tex]

 

[Mandelbroth]

Hello,

I read from several sources the statement that the set of points M[itex]\in[/itex]ℝ² given by [itex](t, \, |t|^2)[/itex] is an example of differentiable manifold of class C¹ but not C².

Is that true?
To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should be able to find an atlas of charts [itex]x_i:U_i \rightarrow M[/itex] such that all the points of M are covered by the atlas.
So how do we cover the point (0,0) ?

Do we need to use 3 charts as follows?

[tex]x_1(t) = (t,\, t^2) \quad \, t\in(0,+\infty)[/tex]
[tex]x_2(t) = (t,\, -t^2) \quad \, t\in(-\infty, 0)[/tex]
[tex]x_3(t) = (t,t^2) \quad \, t\in(-1,1)[/tex]

I don't understand what you're doing. For real numbers ##t##, ##|t|^2=t^2##, right?

 

[mnb96]

Sorry, I reported the wrong equation for the curve . What I meant was that the points of M are given by [itex] (t, \, \mathrm{sign}(t)t^2)[/itex]

 

[WWGD]

Why don't you use a ##C^1## function f that is not ##C^2## and try to find a manifold whose transition functions ae given by f ?

Only one I can think of now is $$f(x)= \frac {-x^2}{2}; x\leq 0$$ and

$$f(x) =\frac{x^2}{2} ; x>0$$

We have f'(x)=|x| , which is not differentiable. I'm not sure this will work; just an idea. Seems like an interesting question: given a (finite) collection of functions, can I find a manifold for which the transition functions are precisely those functions? I'm thinking this is the way one can construct bundles by choosing cocycles satisfying certain properties; can we do something similar for manifolds?

 

[mnb96]

Hi WWGD,

I think the function you suggested is the same as the one I am considering, up to a scalar factor of 1/2.
In my first post I gave three coordinate charts x₁, x₂, x₃ that I regarded as possibly correct candidates to form an atlas.

What I missed were the transition maps.
In this case, it seems that the transition maps are always the identity function on some domain: [tex]\tau_{1,3}(t)=\tau_{3,1}(t)=t \quad ; \, t\in(0,1)[/tex]
[tex]\tau_{2,3}(t)=\tau_{3,2}(t)=t \quad ; \, t\in(-1,0)[/tex]

If this last step is correct, it should prove that the curve is indeed a [itex]C^1[/itex]-differentiable manifold (but not [itex]C^2[/itex]).