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I am trying to learn differential geometry (i.e., teach myself!) So here is a question that came up.

For some $h > 0$, consider the cone

$C_h = \{ (x,y,z) \; : \; 0 \le z = \sqrt{x^2 + y^2} < h \} \subset \mathbb{R}^3$

endowed with subspace topology. It seems that we can cover this with a single chart $(U,\phi)$ where $U = C_h$ and $\phi$ is the projection $\phi(x,y,z) = (x,y)$. So it seems that this defines a differentiable structure and we get a smooth ($C^\infty$) 2-dimensional manifold. (Is it correct?)

Now consider the inclusion map $i : C_h \to \mathbb{R}^3$, is this maps smooth? It doesn't seem to me that it is. The expression of $i$ in the chart above is not smooth at $(0,0)$ and I don't seem to be able to find any other compatible chart around zero which has a smooth representation. (Haven't given it much thought though). If this is true how one shows that this map is not smooth. (Also, if this is true, a vague question is whether removing the origin is the only way to fix this problem)

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does smoothness depend on the chart? –  roy smith Feb 17 '11 at 21:15
This question might get a better answer at math.stackexchange.com. Roughly speaking, it's at undergraduate level, and MO is mostly for graduate-level-and-above questions. For this reason, I'm going to vote to close. –  HJRW Feb 17 '11 at 21:27
I agree with Henry –  Steven Gubkin Feb 17 '11 at 21:33
Okay. Fair enough. I should thank you, Roy, for the hint. I guess you are implying that since smoothness does not depend on the chart, I have shown that the inclusion map is not smooth. Intuitively there is something non-smooth about that point of the pointed cone. I just wanted to confirm that it can be made into a smooth manifold as above (which if true is odd to me and interesting!) and that what is wrong shows itself for example as the non-smoothness of the inclusion map into R^3. (I was also wondering if it is possible to remedy this somehow or is this in some sense intrinsic.) –  passerby51 Feb 17 '11 at 22:33
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closed as too localized by Andres Caicedo, HJRW, Mariano Suárez-Alvarez, Qiaochu Yuan, Steven Gubkin Feb 17 '11 at 21:33

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