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I am aware of two conventions for the Riemann curvature tensor, namely the expression $$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$ is either declared to be $R(X,Y,Z,W)$ or $R(W,Z,X,Y)$. These differ by a sign. A reason to prefer the first convention is that the order of $X,Y,Z,W$ is preserved. A reason to prefer the second convention is that the expression for the sectional curvature makes more sense: $R(a,b,a,b)$ where $a$ and $b$ form an orthonormal basis of a plane inside the tangent space. Are there other reasons to prefer one of these conventions over the other? What is the history of these two conventions? Why these and not any other orderings of $X,Y,Z,W$ (the second convention has a somewhat bizarre ordering, so I wonder if there's a specific reason for it)?

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  • $\begingroup$ Wild guess: the slots for X and Y are kept next to each other as either the first two or the final two due to the obvious antisymmetry between those two slots. The pairwise symmetries (X,Y) <-> (Z,W) of the Riemann tensor means we want to place (X,Y) either in the first two or the final two slots to emphasize this symmetry. Finally, before the use of invariant notation when everyone uses coordinate indices, it is easier to remember the meaning of $R^i_{jkl}$ if $i$ is the "extreme" index (first most or final most). $\endgroup$
    – Willie Wong
    Commented Sep 23, 2019 at 16:59
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    $\begingroup$ Note also, that O'Neill, for example, writes $$ R(X,Y)Z = \nabla_{[X,Y]} Z - [\nabla_X, \nabla_Y] Z $$ which is in some sense an ordering that you didn't list. $\endgroup$
    – Willie Wong
    Commented Sep 23, 2019 at 17:01
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    $\begingroup$ I prefer strongly defining the operator $R(X,Y) = [\nabla_X, \nabla_Y] - \nabla_{[X,Y]}$ mentioned by @WillieWong, and writing $\langle W, R(X,Y)Z\rangle$. It removes all ambiguity from the order of the vector fields. If I want to write the components of the tensor with respect to an orthonormal basis, then I write $R(e_k,e_l)e_j = R^i{}_{jkl}e_i$, and $e_i\cdot R(e_k,e_l)e_j = R_{ijkl}$. The reason I like this is that the skew symmetric pair of indices matches the pair of vector fields $X$ and $Y$ with respect to which the curvature tensor is explicitly skew symmetric. $\endgroup$
    – Deane Yang
    Commented Sep 23, 2019 at 21:18
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    $\begingroup$ On the other hand, it is all too easy to forget what exactly $R(X,Y,Z,W)$ means. $\endgroup$
    – Deane Yang
    Commented Sep 23, 2019 at 21:20
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    $\begingroup$ @DeaneYang $R(e_k,e_l)e_j = R^i_{jkl}e_i$ is also the convention for indices of Riemann tensor preferred in Physics. $R(e_k,e_l)$ is a linear operator in the tangent space, so we put indices k and l together as a pair, and indices i and j as the matrix indices for the linear operator. $\endgroup$
    – liyiontheway
    Commented May 8 at 12:11

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