Riemann curvature tensor

Remarks

$R$ is $C^\infty(M)$-linear in $X$, $Y$ and $Z$, so it is a tensor field of type $(1, 3)$, and $R(Y, X) = -R(X, Y)$. Some books use the opposite sign; the convention here gives positive sectional curvature $g(R(X, Y) Y, X) > 0$ on the sphere. Example: for the flat connection of $\R^n$, $\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z = \nabla_{[X, Y]} Z$ by symmetry of second derivatives, so $R = 0$. For the round sphere of radius $r$, $R(X, Y) Z = r^{-2} \big( g(Y, Z) X - g(X, Z) Y \big)$. Non-example of a tensor: the expression $\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z$ alone is not $C^\infty(M)$-linear in $X$, as replacing $X$ by $fX$ adds $-(Yf) \nabla_X Z$; the term $\nabla_{[X, Y]} Z$ removes this. In a chart, $R(\partial_i, \partial_j) \partial_k = \sum_l R^l{}_{kij} \partial_l$ with $R^l{}_{kij} = \partial_i \Gamma^l_{jk} - \partial_j \Gamma^l_{ik} + \sum_m (\Gamma^l_{im} \Gamma^m_{jk} - \Gamma^l_{jm} \Gamma^m_{ik})$. For the Levi-Civita connection, $R_{abcd} = g(R(X_c, X_d) X_b, X_a)$ is antisymmetric in $(a, b)$ and in $(c, d)$ and symmetric under $(a, b) \leftrightarrow (c, d)$.