Is the finding trace of the Riemann tensor the same thing as contracting two indices?

Question

To form the Ricci curvature tensor, we have to take the trace of the Riemann tensor. But I also know \begin{equation} R_{ij} := R_{kij}^{\phantom{kij}k} \end{equation}

Can someone show me why taking the trace is the same thing as contracting indices?

Answer 1

Consider the following matrix

\begin{equation} A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} \end{equation}

By definition, the trace of $A$ is defined as

\begin{equation} \text{tr}(A) = a + e + i = a_{11} + a_{22} + a_{33} \end{equation}

So think about how we write tensor indices. If I have a tensor $a_{ij}$ and I contract one index to get $a_i^i$, now both indices have the same value. So for all values of $i$ in this case we would have $a_1^1$, $a_2^2$, $a_3^3$, which corresponds to $a$,$e$,$i$.

Thus, contracting indices gives the same value as taking the trace.