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If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres. If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R \sin(\frac{r}{R})$, where $R$ is the radius of the 2-Sphere. Similarly for 3D positively curved space the radius of the spheres at a distance $r$ from the origin becomes $R \sin(\frac{r}{R})$,in this case whether $R$ is the radius of the 3-spheres?

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  • $\begingroup$ If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on. $\endgroup$ – John Rennie Jan 2 at 8:14
  • $\begingroup$ Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere?? $\endgroup$ – Apashanka Das Jan 2 at 9:25
  • $\begingroup$ If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question. $\endgroup$ – John Rennie Jan 2 at 9:46
  • $\begingroup$ R is the radius of curvature $\endgroup$ – Reign Jan 2 at 15:44
  • $\begingroup$ For 3-D case is it the radius of the 3-sphere?? $\endgroup$ – Apashanka Das Jan 2 at 15:57
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For 3-Sphere the radius of the Curvature is R. For more information look this Cosmology Mathematical Tripos

If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.

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