Why isn't the graph of $y=|x|$ a smooth manifold?

Asked Oct 09 '13 at 15:41

Active Oct 09 '13 at 15:41

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Consider the graph of $y=|x|$ from $-1<x<1$. Equip it with a single chart, the projection onto the $x$-axis. Is it now a smooth manifold? It seems like it shouldn't be smooth, but perhaps with only one chart, any manifold is smooth (because it trivially satisfies the compatibility condition)?

What chart could we put on there that would not be smoothly compatible with this first one?

asked Oct 09 '13 at 15:41

Eric Auld

28,127

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It is a one-dimensional smooth manifold. – njguliyev Oct 09 '13 at 15:44
Think it this way, the "non smothness" of $y=|x|$ is on the way it is sitting in $\mathbb{R}^2$ not in the graph itself which is just $\mathbb{R}$. – Martin Citoler Oct 09 '13 at 15:47

Why isn't the graph of $y=|x|$ a smooth manifold?

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