I Is the FLRW Metric the Only Form for One-Dimensional Maximally Symmetric Spaces?

[davidge]

If you are asking, does a maximally symmetric space have to have constant curvature, see if you can deduce the answer from the definition of maximally symmetric space. (Hint: Carroll's discussion in Chapter 8 of his notes is helpful here.)

I think the answer is "yes", not deduced from Carroll's notes, but I remember seen this discussion on Weinberg's book.

If I remember well, constant curvature is a consequence of a maximally symmetric space.

 

[davidge]

constant curvature

In the case in question (FLRW metric), this would be so for any value of $k$. But then would this imply that even a non homogeneous space (case $k=0$) has maximal symmetry?

 

[PeterDonis]

I think the answer is "yes"

I agree.

In the case in question (FLRW metric), this would be so for any value of $k$

Yes.

would this imply that even a non homogeneous space (case $k=0$) has maximal symmetry?

The $k = 0$ case is just Euclidean 3-space; it should be obvious that this space is maximally symmetric. I don't know why you think this case is non-homogeneous.