# Intuitively what's the difference between Lie Derivative and Covariant Derivative?

Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...

cristo
Staff Emeritus
garrett
Gold Member
The covariant derivative uses a connection, while the Lie derivative doesn't.

Just to spotlight one of these: The Lie derivative L_X(Y) is basically a dynamical invariant. It measures how compatible the flows of the vector fields are, i.e. how much they commute with each other. If X is generated by the flow $\phi_t$ and Y is generated by $\psi_s$, then the Lie derivative [X,Y] at point P is the tangent vector at time 0 of the curve given by: $t \mapsto \psi_{-t} \circ \phi_{-t} \circ \psi_t \circ \phi_t (P)$.

Berger describes this as moving P forward in time along the X-curves by t, then moving along the Y-curve by t, then moving backward in time along the X-curve and finally moving backward in time along the Y-curve. If you've ended up back at P, then [X,Y]=0 at P.

In particular, if $\phi_s \circ \psi_t=\psi_t \circ \phi_s$ for all s and t, then [X,Y]=0.

The covariant derivative acts similarly except instead of pushing Y along the X-curves via X's flow, we are pushing Y along X-curve via parallel transport. This explanation, though, is a bit of circular logic, since one usually uses the specific covariant derivative to generate the parallel transport.

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