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What is the metric (distance function) in S1 or S2 which is the 1 sphere and 2sphere respectively.

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- #1

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What is the metric (distance function) in S1 or S2 which is the 1 sphere and 2sphere respectively.

- #2

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In general, if J is the jacobian matrix of the manifold:

[itex]\mathbf{g} = J^T J[/itex]

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HallsofIvy

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For the 2-sphere (the surface of the unit sphere), we can take [itex]x= cos(\theta)sin(\phi)[/itex], [itex]y= sin(\theta)sin(\phi)[/itex], [itex]z= cos(\phi)[/itex] (spherical coordinates with [itex]\rho= 1[/itex]). Then [itex]dx= -sin(\theta)sin(\phi)+ cos(\theta)cos(\phi)[/itex], [itex]dy= cos(\theta)sin(\phi)+ sin(\theta)cos(\phi)[/itex], and [itex]dz= -sin(\phi)d\phi[/itex].

[tex]dx^2= sin^2(\theta)sin^2(\phi)d\theta^2- 2sin(\theta)cos(\theta)sin(\phi)cos(\phi)d\theta d\phi+ cos^2(\theta)cos^2(\phi)d\phi^2[/tex]

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