If we define the Ricci curvature for an orthonormal frame, we can simplify the sectional curvature formula to get:
$$Ric(v, v) = \sum R(e_i, v)v \cdot e_i$$
However, I obtain a different formula to the one that is ordinarily presented. Supposing we replace $v$ for its components and use the linearity of the Riemann curvature tensor, we can write:
$$Ric(v, v) = \sum R(e_i, v^je_j)v^ke_k \cdot e_i$$
$$Ric(v, v) = \sum v^j v^k R(e_i, e_j)e_k \cdot e_i$$
$$Ric(v, v) = \sum v^j v^k R^m_{kij}e_m \cdot e_i$$
But this is the point where I hit a bit of trouble. Shouldn't it be possible to replace $e_m \cdot e_i$ for the metric tensor components $g_{mi}$, which would give me:
$$Ric(v, v) = \sum v^j v^k R^m_{kij}g_{mi}$$
$$Ric(v, v) = \sum v^j v^k R_{ikij}$$
At this stage, I'd define:
$$R_{kj} = R_{ikij}$$
But that seems awkward because usually we'd:
- Contract over an upper and lower index (and)
- The usual definition has $R_{kj} = R^i_{kij}$
So I'd like to figure out where the logic I've applied is wrong here
Best Answer
This upper/lower indice confusion arises when you choose an orthonormal frame, since in this case $g_{ij} = \delta_{ij}$.
The definition of the Ricci curvature is $ R_{kj} = R^i_{kij}$, where we used the Einstein summation convention. To be precise it's $$ R_{kj} = \sum_i R^i_{kij}, $$ then by definition $$ R_{kj} = \sum_i R^i_{kij} = \sum_{i,l} g^{li} R_{lkij}. $$ Using orthonormal frame, we have $$ \tag{1} R_{kj}= \sum_{i,l} \delta_{il} R_{lkij} = \sum_{i} R_{ikij}. $$
If by $R_{kj} = R_{ikij}$ you mean (1), then there is no mistake, but bad writing style: you used Einstein summation notation even for $R_{ikij}$, which is not invariant under change of (arbitrary) coordinates.
The practice is that we used Einstein notation only for tensor. When you want to calculate using a specific frame, we will be honest and write down all the summations. Sometimes we even use different indices ($\alpha, \beta, \gamma$) to distinguish it from the indices for coordinates basis ($i, j, k$).