Solution
= Solution
Fix the convention
$$
(\nabla_c\nabla_d-\nabla_d\nabla_c)X^a=R^a{}_{bcd}X^b.
$$
Equivalently,
$$
R(U,V)X=\nabla_U\nabla_VX-\nabla_V\nabla_UX-\nabla_{[U,V]}X.
$$
This defines the <Riemann curvature tensor> because it is $C^\infty(M)$-linear in $U,V,X$: derivatives of a multiplying function cancel between the three terms. Thus its value at a point depends only on the three tangent vectors there, rather than their extensions.