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This other question exists, but it doesn't answer my question: Geometric interpretation of the second covariant derivative

I know the Riemann Tensor can be written as the commutator of the second covariant derivative (assuming the connection is torsion-free):

$$R(u,v)w = \nabla_{u,v}^2 w - \nabla_{v,u}^2 w$$

where $\nabla_{u,v}^2 w = \nabla_u \nabla_v w - \nabla_{\nabla_u v}w$ is the "second covariant derivative".

What I'm missing here is the "why". I don't understand the geometrical meaning of $\nabla_{u,v}^2 w$, or why anyone bothered to invent this expression. How was the formula for the "second covariant derivative" invented, and what is its meaning?

I tried drawing some diagrams showing how the vectors are positioned, but they did not bring me much insight. The other question has better diagrams in the answers.

Here is a visualization of $\nabla_u \nabla_v w$:

enter image description here

Here is a visualization of $\nabla_{\nabla_u v}w$:

enter image description here

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  • $\begingroup$ I'm struggling with similar questions. Did you find your answer or any explanation to this question? $\endgroup$
    – Shirish Kulhari
    Feb 14 at 21:03
  • $\begingroup$ @ShirishKulhari I did not. Instead I learned to accept the definition of the Riemann Tensor as $R(u,v)w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]} w$, where the 3rd term takes into account that trying to form a parallelogram in curved space leads to a "gap" that must be closed. The Lie Bracket term $[u,v]$ represents this "5th" side of the loop that must be taken into account when parallel transported a vector around a loop. $\endgroup$
    – eigenchris
    Feb 16 at 5:06

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