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Since Riemann with 1 upper and 3 lower index mean the changes of a vector after being parallel transported around a closed loop, does Riemann with all 4 lower indexes mean the changes of a co-vector after being parallel transported around a closed loop? If not, what does it actually mean?

Note that, What is the physical meaning of the connection and the curvature tensor? does not answer my question. I know what $R^a_{bcd}$ means, I also understand the meaning of the Ricci Tensor. What I'm curious about is the meaning of Riemann in all indeces lowered, i.e, $R_{abcd}$.

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