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Reference: here

Given 3-dim Euclidean metric in spherical coordinates by

$$ds^2 = dr^2 + r^2 d\theta^2 + r^2 \text{sin}^2 \theta d\phi^2 \tag{1}$$

so restricting to

$$r=R=const. \tag{2}$$

where $R$ is scalar Ricci, gives

$$ds^2 = R^2 d\theta^2 + R^2 \text{sin}^2 \theta d\phi^2 \tag{3}$$

Questions:

  1. What is the physical meaning of $r=R=const.$ as stated in $(2)$?

  2. What is the relation between $r$ and $R$?

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The metric you present uses curvilinear coordinates for flat (non-curved) R$^3$ (Euclidean 3-d space).

When you restrict to r = constant, the resulting spherical surface is 2-dimensional and curved.

For any curved space (or space-time) you can go through often lengthy calculations to compute the Ricci scalar, R. For a spherical surface, R happens to have the same value as the radius of the sphere, r.

On a 2-d surface, the scalar R tells us how much a gyroscope (vector that keeps pointing "in one direction" turns when you move in a small loop. In higher dimensions, the tensor R$^a_{bcd}$ hold more information, as loops in different directions and differently pointing vectors, end up turning by different amounts. R does then not have have this simple interpretation, but still plays a crucial role in Einstein's equations, relating curvature to mass.

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