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For quite some time I've been wondering now whether there's a well-understandable difference between the Lie and the covariant derivative. Although they're defined in fundamentally different ways, they're both (in a special case, at least) standing for the directional derivative of a vector. However, the Lie derivative doesn't make any assumptions about whether there's a connection on the manifold or not.

So where's the difference in between the two derivatives, where's the structure of the manifold playing into what the result of the derivative will be?

With best regards,

Quchen