§ Lie derivative versus covariant derivative

• Lie derivative cares about all flow lines, covariant derivative cares about a single flow line.
• The black vector field is X
• The red vector field $Y$ such that $L_X Y = 0$. See that the length of the red vectors are compressed as we go towards the right, since the lie derivative measures how our "rectangles fail to commute". Thus, for the rectangle to commute, we first (a) need a rectangle, meaning we need to care about at least two flows in $X$, and (b) the flows (plural) of $X$ force the vector field $Y$to shrink.
• The blue vector field $Z$ is such that $\nabla_X Z = 0$. See that this only cares about a single line. Thus to conserve the vectors, it needs the support of a metric (ie, to keep perpendiculars perpendicular).

• Reference question