Surface area of a 2-sphere in Abstract Index Notation 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.
 A: 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.)
