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Sphere Embedding


A 4-sphere has positive curvature, with

 R^2=x^2+y^2+z^2+w^2
(1)
 2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)+2w=0.
(2)

Since

 r=xx^^+yy^^+zz^^
(3)
dw=-(xdx+ydy+zdz)/w
(4)
=-(r·dr)/(sqrt(R^2-r^2)).
(5)

To stay on the surface of the sphere,

ds^2=dx^2+dy^2+dz^2+dw^2
(6)
=dx^2+dy^2+dz^2+(r^2dr^2)/(R^2-r^2)
(7)
=dr^2+r^2dOmega^2+(dr^2)/((R^2)/(r^2)-1)
(8)
=dr^2(1+1/((R^2)/(r^2)-1))+r^2dOmega^2
(9)
=dr^2(((R^2)/(r^2))/((R^2)/(r^2)-1))+r^2dOmega^2
(10)
=(dr^2)/(1-(r^2)/(R^2))+r^2dOmega^2.
(11)

With the addition of the so-called expansion parameter, this is the Robertson-Walker line element.


Cite this as:

Weisstein, Eric W. "Sphere Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphereEmbedding.html