Elementary question in differential geometry - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-18T05:41:14Z http://mathoverflow.net/feeds/question/55783 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55783/elementary-question-in-differential-geometry Elementary question in differential geometry passerby51 2011-02-17T20:56:16Z 2011-02-17T20:56:16Z <p>I am trying to learn differential geometry (i.e., teach myself!) So here is a question that came up.</p> <p>For some $h > 0$, consider the cone </p> <p>$C_h = \{ (x,y,z) \; : \; 0 \le z = \sqrt{x^2 + y^2} &lt; h \} \subset \mathbb{R}^3$</p> <p>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?)</p> <p>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)</p>