On page 28 of Hicks' Notes on Differential Geometry he proves:
(9) $D_X D_Y Z - D_Y D_X Z - D_{[X,Y]} Z = \langle L(Y),Z\rangle L(X) - \langle L(X),Z\rangle L(Y)$
where $D_X$ is the induced covariant derivative: $D_X Y= \overline D Y -\langle L(X),Y \rangle N $ and $\overline D_X Y$ is the usual directional derivative.
He then defines the left hand side as R(X,Y)Z and states that "(9) implies thatR(X,Y)Z depends only on $X_p,Y_p,Z_p$". How can I see this if the covariant derivative $D_X$ is defined using the normal vector N?