Dependence of Riemann curvature tensor on the values of the vector fields in a neighborhood.

Question

The Riemann curvature tensor is given by $$R(X, Y)Z= \nabla_X \nabla_Y Z- \nabla_Y \nabla_X Z- \nabla_{[X, Y]} Z$$ for vector fields $X$, $Y$ and $Z$. My question is: The value $\nabla_R S$ in a particular point for vector fields $R$ and $S$ obviously depends on the values of $S$ in a neighborhood of this point. Does the value $(X, Y)Z= \nabla_X \nabla_Y Z- \nabla_Y \nabla_X Z- \nabla_{[X, Y]} Z$ of the Riemann curvature tensor in a particular point only depend on the values of $X$, $Y$ and $Z$ in this point or does it only depend on the values in that point?

Answer 1

The quantity $R(X,Y)Z$ is a tensor, that is: $$R(fX,gY)(hZ) = fghR(X,Y)Z,$$for all functions $f,g,h$, aside from being additive in each entry. So the value of $(R(X,Y)Z)(p) = R_p(X_p,Y_p)Z_p$ depends only of the values of $X,Y$ and $Z$ at $p$, and not on some neighborhood of it.

This holds for arbitrary tensor fields. Look at Proposition $2$ in page $37$ of O'Neill's Semi-Riemannian Geometry with Applications to Relativity, for example.