Surface area of a 2-sphere in Abstract Index Notation

Question

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by dropping into a coordinate system and doing a concrete integral, but it seems all the relevant information is available at the abstract level. However, it is not clear to me what techniques would show this without a concrete coordinate system.

Answer 1

First: The two-sphere is actually not the only surface which admits a metric with constant positive curvature. There's also the real projective plane which is $S^2/\{\pm 1\}$, the two-sphere with antipodal points identified.

Second: There is a nice formula relating the area, the curvature, and precisely which (topological) type of surface you're working with.

Let $\chi(M)$ be the Euler characteristic of $M$. For $M = S^2$ this is $2$, for the same reason as $V-E+F = 2$ for polyhedra. For $M = \mathbb{RP}^2$, this is $1$ (basically because $S^2$ maps two-to-one to it, so if you draw vertices, edges, and faces on it, you'll get twice as many of each back on $S^2$).

Let $K$ be the Gaussian curvature. Here this is $1$ for the curvature tensor you describe.

Then the Gauss-Bonnet theorem states that $$\int_M K dA = 2 \pi \chi(M)$$

Thus in your case the LHS is $\int_M 1\, dA = \mathrm{Area}(M)$ and the RHS is $2\pi\chi(M)$ which is $4\pi$ for the sphere and $2\pi$ for $\mathbb{RP}^2$.

(There are variants of this for surfaces with boundary and for manifolds in higher dimensions.)