I Expansion of 3-D positively curved space

[Apashanka]

The metric of a 3-D positively curved space is dr²+ S_k(r)²(dθ²+sin²θdΦ²).
Now if this space expands with a scale factor a(t) from r to r'.
Whether the change in the radial component be a(t)dr and angular component be S_k(r')dθ and S_k(r')sinθdΦ since the change due to expansion is already incorporated in S_k(r')??

 

[Orodruin]

since thethe cha due to expansion is already incorporated in Sk(r')??

It is not incorporated in $S_k(r)$.

 

[Apashanka]

It is not incorporated in $S_k(r)$.

If we imagine 3-D euclidean space consists of infinite no. of concentric spheres,after given positive curvature the radius of the spheres become Rsin(r/R)=S_k(r),as opposed to r for flat 3-D space (where R is radius of curvature) where r is the distance from the origin .

Now if this sphere of radius S_k(r) expands from r to r' having scale factor a(t) in time t.
Then the elementary length on the surface of sphere will change only due to the factor that it's radius got changed from S_k(r) to S_k(r') in time t,scale factor effect will not be on this.
Am I right??

 

[Orodruin]

Now if this sphere of radius Sk(r) expands from r to r' having scale factor a(t) in time t.

Sorry, but this makes no sense. $r$ is a coordinate, it is $R$ that is the radius of curvature. Either way, the standard RW coordinates have a fixed $R$ and $r$ is also fixed for a comoving observer. The spatial part of the RW metric is of the form
$$
d\Sigma^2 = a(t)^2 [dr^2 + S_k(r)^2 d\Omega^2].
$$
Now, you could introduce a new coordinate $r' = a(t) r$ such that the spatial part of the RW metric takes the form
$$
d\Sigma^2 = dr'^2 + [a(t) S_k(r)]^2 d\Omega^2 = dr'^2 + \tilde S_k(r',t)^2 d\Omega^2,
$$
where
$$
\tilde S_k(r',t) = a(t) S_k(r) = a(t) R \sin\left(\frac{r'}{a(t) R}\right) = R(t) \sin(r'/R(t))
$$
and $R(t) = a(t) R$. However, your $r'$ will then no longer be a comoving coordinate and you will introduce cross terms between the spatial coordinate $r'$ and the time coordinate $t$ in the metric. This is highly non-recommended.

 

[Apashanka]

r is also fixed for a comoving observer.

Sir for the 2-D positively curved plane the radius of the circles become Rsin(r/R) instead of r ,at a distance r from the origin where R is the radius of curvature or radius of the 2-sphere and it can be drawn geometrically.
Similarly for 3-D positively curved space the radius of the spheres become Rsin(r/R) instead of r, at a distance r from the origin.
Sir for the 2-D case the R is the radius of 2-Sphere and can be visualised but sir for 3-D case what R is actually here??is there any method to visualise this mathematically??