A cone is a topologic manofold but can we define a differentiable manifold structure on it?
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1What is a cone? – Lee Mosher Feb 28 '16 at 22:10
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Z=radical(x^2+y^2) – user297564 Feb 28 '16 at 22:21
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The projection $p$ from the cone onto the $x,y$-plane is a homeomorphism. So you can use $p$ to pull back the differential manifold structure on the $x,y$-plane and you get a differential manifold structure on the cone.
Lee Mosher
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Is a curve $c$ on the cone smooth iff $p \circ c$ is smooth in $\Bbb R ^2$? If so, then this is weird: there are "V"-shaped curves on the cone that are "visually" not smooth (they have a pointed vertex at the tip of the cone), the projection of which is smooth. I can't understand this visual paradox. – Alex M. Feb 28 '16 at 22:29
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@AlexM.: The cone can be given a smooth structure when considered as an abstract topological manifold, but that structure is not straightforwardly obtainable from its embedding in $\Bbb{R}^3$. (As you've noticed, it has "too many" smooth curves. Conversely, it has "too few" smooth functions; e.g., the restriction of $z$ to the cone is not a smooth function.) – Micah Feb 28 '16 at 22:48