I'm studying de Sitter manifolds and am confused about the difference between the choice of foliation and the choice of coordinates (and how they relate to the spatial curvature). I can choose the natural coordinates with $K = 1$, and this makes the spatial slices surfaces of 3-spheres. On the other hand, I can choose conformally flat coordinates with $K = 0$ and I assume the spatial slices are sheets here. Still, in both cases the metric has the form (neglecting the pre-factor)
$$ds^2 = -d\eta^2+d\Sigma^2$$
and I can choose either
$$d\Sigma^2 = dx^2 + dy^2 + dz^2$$
or
$$d\Sigma^2 = dr^2 + r^2d\Omega_2^2$$
which I visualize as sheets and balls respectively. I understand de Sitter is a special case due to its many symmetries, but this is confusing me about the difference between the spatial curvature ($K$), the foliation, and the labeling (i.e. coordinates).
Thanks in advance for any help!