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In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - F^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether space is positively or negatively curved. This is a simplified version of $K$, which is the Gaussian curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.

In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - F^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether spacetime is positively or negatively curved. This is a simplified version of $K$, which is the Gaussian curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.

In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - F^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether space is positively or negatively curved. This is a simplified version of $K$, which is the spatial curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.

In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - F^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether space is positively or negatively curved. This is a simplified version of $K$, which is the Gaussian curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.

In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - R^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether space is positively or negatively curved. This is a simplified version of $K$, which is the spatial curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.

In the Robertson-Walker metric,

$$ (\text{ds})^2 = c^2(\text{dt})^2 - F^2(t)\left[ \frac{(\text{dr})^2}{1-kr^2} + r^2(\text{d}\theta)^2 + r^2\sin^2\theta (\text{d}\phi)^2 \right] $$

The $k$ in the second term is the curvature parameter that takes the values +1 or -1 depending on whether space is positively or negatively curved. This is a simplified version of $K$, which is the spatial curvature. The spatial curvature is then related to the Ricci scalar $R$ which is a number that is determined by the geometry of the space around it.