Wikipedia has the following formula for the curvature of a surface:
$$K = \lim_{r\to 0}12 \frac{\pi r^2 - A(r)}{\pi r^4}$$ where $A(r)$ is the area of a geodesic disk.
Now, in this paper, in Theorem $3.1$, page $337-338$, they have an approximation for the volume of a $n-ball$ of radius $r$. In the case $n=2$, their approximation: $$V(r) = \pi r^2\bigg(1 - \frac{\tau(R)}{24}r^2 + O(r^4)\bigg)$$ where $R$ is the curvature tensor and $\tau(R) = \sum_i R_{ii}$ and $R_{ij} = \sum_k R_{ijik}$ as defined on page $333$ of the paper.
However, isn't $R_{ii}$ as defined always zero? In general, we have $R_{ijkl} + R_{jikl} = 0$ and therefore $2R_{iikl} = 0$ so that $R_{ii} = \tau(R) = 0$? This would seem to imply that $K = 0$ identically for any surface!
Where did I go wrong?
No, $R_{ij} = \sum R_{ikjk}$ is the Ricci tensor here. You've got a typo up there or else they have one in the paper. (Of course, for a surface, Ricci curvature is just the scalar (Gaussian) curvature.)