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2

So the function defined by $$f(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi, \cos\theta)$$ is not a coordinate chart but rather an embedding $\mathbb{S}^2\hookrightarrow \mathbb{R}^3$. The reason it cannot be a coordinate chart is as follows: A coordinate chart must be a homeomorphism from the open sets $\mathcal{U}_i$ of the manifold $M$ to ...

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Can these two pictures be connected in some way? Yes, that's why the Wikipedia spinor article features a picture of a Möbius strip: GNUFDL image by Slawekb, see Wikipedia The Mobius strip also features in the Mathspages Dirac's belt article where you can read that it's "reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be ...

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The simple answer to this is "yes, go read the paper". Indeed, the OP is somewhat negligent in their post, since the Celi et al. paper plainly states in the introduction that We also show that by using additional Raman and radio-frequency transitions one can connect the edges in the extra dimension. Thus, the possibility of engineering synthetic ...

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