Deriving the line element in homogenous isotropic space

1. Feb 26, 2008

TheMan112

If the Ricci-scalar $$R$$ is constant for a given spatial hypersurface, then the curvature of that region should be homogenous and isotropic, right?

A homogenous and isotropic hypersurface (disregarding time) has by definition the following line element (due to spherical symmetry):

$$d\sigma^2 = a^2 \left(\frac{1}{1-kr^2} dr^2 + r^2(d \theta^2 + sin^2(\theta) d \Phi^2) \right)$$

Where k = -1, 0 or +1 and a is constant.

Why $$\frac{1}{1-kr^2} dr^2$$ ?

This is apparently very important as the value of k determines the evolution of the universe, but I don't know how to come to this line element.

2. Feb 26, 2008