From: ask@ucscb.UCSC.EDU (Andrew Stanford Klingler)
Newsgroups: sci.math
Subject: Re: Lie vs. Covariant derivatives
Date: 16 Nov 92 09:24:44 GMT
Distribution: usa
Organization: University of California; Santa Cruz
NNTP-Posting-Host: ucscb.ucsc.edu


:From: glezen@drb-mathsun1.usc.edu (Paul Glezen)

:   I've seen several interesting responses to the inquiry: 

:   "What is the difference between covariant derivative and the Lie
:    derivative?"

:   But comments like "one requires a certain vector field to be defined at
:a point while the other needs it to be defined on some neighborhood", while
:correct, shed little light on why you get different answers.  Perhaps a simple
:example will show why these things are two completely different beasts.  The
:emphasis here is on simplicity and intuition; not technicalities meaningless
:to a beginner.

:   Let's choose our manifold to be the 2-d plane and let P be the point (1,0).
:Consider two vector fields V and W defined by

:      V = (-y)i + (x)j        where i and j are the standard basis for
: and                          the plane and hence for the tangent space
:      W =  i                  to the plane.

:   V corresponds to the velocity field arising from a rigid counter-clockwise
:rotation about the origin while W is the constant vector field which points to 
:the right and has length one.  Allow me to denote

:      L(V,W)  := Lie derivative of W in the direction of V

:      D(V,W)  := Covariant derivative of W in the direction of V

:   Notice also the diffence between L and D when you switch V and W.  Since it
:can be shown that L(V,W) = [V,W], the Lie bracket, we expect L(V,W) = -L(W,V).
:In our example, it is clear that the diffeomorphisms induced by W are merely
:translations to the right.  The tangent mappings for each diffeomorphism is
:clearly the identity.  If you start at (1,0) and move to the right, indeed 
:the vectors of V, the rotation field, seem to get larger in the j direction.
:Thus when you compare V(1+dx,0), via the identity tangent map of the diffeo-
:morphism induced by W, to V(1,0), you see that indeed the vector field V is
:getting bigger in the j direction.  Thus L(W,V) = j.

:   On the otherhand, it is easy to see that D(W,V) = -i at the point P if you
:think of V as being the velocity field of a particle rotating about the origin
:at constant speed.  Then D(W,V) is just the acceleration (change in velocity)
:which points toward the origin.

D(W,V) is the covariant derivative of V in the direction of W.  But
W is a constant i pointing along the x-axis.  An infinitesimal 
displacement along the x-axis causes V to change by (dx)j, so
we get D(W,V)=j.  This is merely a calculational error, but it brings
back to mind the equation that underlay my geometric confusion.

L(V,W) = [V,W] = D(V,W) - D(W,V)

So in some restricted sense I can think of the Lie derivative as
constructed out of covariant differentiation around a little loop
(incidentally this makes antisymmetry of Lie derivative obvious).
This is what I was mentally referring to with my previous post.
However, the equation's only true for torsion-free connections.

As a followup: what does the existence of torsion mean to those
of us whose experience is limited to GR? (Apart from a bad choice
of frame field!)

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