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

1. Feb 5, 2007

### sit.think.solve

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

2. Feb 5, 2007

### cristo

Staff Emeritus
3. Feb 5, 2007

### garrett

The covariant derivative uses a connection, while the Lie derivative doesn't.

4. Feb 6, 2007

### Doodle Bob

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.

Last edited: Feb 6, 2007