Hello, Your forum was very helpful in giving intuitive geometric explanations to various concepts from differential geometry, so I was hoping you could perhaps help me with the following. I'm interested in Killing vector fields. I understand that a tangent vector field X is a Killing vector field if the Lie derivative of the metric with respect to X is 0 - this has some geometric meaning. On the other hand, the Killing equation is specified in terms of the covariant derivative of X, and one reaches this equation by manipulating the Lie derivative condition. My question is: is there some intuitive (geometric) explanation to the Killing equation? Thank you.