- #36

lavinia

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fresh_42 said:So you consider an atlas of ##M=\{x^3\}## with one chart ##\mathbb{R}## and two chart mappings ##\varphi## and ##\psi##. These should be functions from ##M## to ##\mathbb{R}##. Say we define ##\varphi(p)=x## and ##\psi(p)=x^3## if ##p=x^3## and the question is, whether they both can be put in one atlas. Is this correct?

Edit: In this case the overlaps are ##\varphi \psi^{-1}(x)=\varphi(x)=x## and ##\psi \varphi^{-1}(x)=\psi(x^3)=x^3## which are both smooth as it should be.

Yes. In the manifold ##(\mathbb R, x^{3})## all charts must be compatible with ##φ(x) = x^3##. The differentiable structure is a maximal atlas of mutually compatible charts. Compatible just means that the transition functions are smooth. https://en.wikipedia.org/wiki/Atlas_(topology)#Transition_maps

https://en.wikipedia.org/wiki/Smooth_structure

For any open set ##U⊂(\mathbb R, x^3)## the map ##ψ(x) = x## is not compatible with ##φ(x) = x^3## since if it were, the transition function ##ψ \circ φ^{-1}## would have to be a smooth map from ##φ(U)⊂ \mathbb{R}## into ##\mathbb{R}##. But ##ψ \circ φ^{-1}(t) = ψ(t^{1/3}) = t^{1/3}## which is not even differentiable at zero.

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