- #1
mnb96
- 715
- 5
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 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]
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