An intuitive explanation to the Killing equation?

kvf

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.

bndnchrs

I think the easiest way to explain it is by what Wikipedia has:

A Killing field is one where when you move points along the field, distances are preserved.

So distances are not distorted when you'e got a Killing field.

kvf

Thank you bndnchrs, but I was looking for an intuitive explanation as to why the Killing equation, given in terms of the covariant derivative, implies that the deformation generated by a Killing vector field preserves distances.

For example, the definition using the Lie derivative (the Lie derivative of the metric = 0), means the metric does not change, and so the deformation preserves distances.
But why does the Killing equation imply the deformation preserves distances?