From: anderson@fermi.phys.ualberta.ca (Warren G. Anderson)
Subject: Re: Covariant vs. Lie Derivative in Gen. Rel.?
Organization: University Of Alberta, Edmonton Canada
Date: Thu, 12 Nov 1992 17:27:48 GMT

In article <1992Nov11.062853.22717@galois.mit.edu> jbaez@riesz.mit.edu (John C.  
Baez) writes:
> In article <1992Nov10.173224.6690@vision.ummed.edu> lml@vision.ummed.edu  
(Lawrence Mark Lifshitz) writes:
> 
> >So, conceptually, what is the difference between the two?
> >Does one require a different type of topogical space or
> >parameterization?  Mathematically how do they differ?
> >thanks in advance.
> 
> Here are the big differences:
> 
> 1) To perform a Lie derivative one needs, not just a tangent vector at a
> point of the manifold, but a whole vector *field* - at least on some
> patch (open set) of the manifold.  This is because one needs to flow the
> tensor along the vector field, so to speak, and see at what rate it
> changes as it flows past a given point.
> 
> 2) The covariant derivative only requires a tangent vector at one point
> of the manifold.  The price you pay is this: to define it you need to
> choose a connection on the (tangent bundle of) the manifold.  Of course,
> such a connection - the Levi-Civita connection - comes for free if your
> manifold has a Riemann metric on it.  

Or even a pseudo-Riemannian metric. In fact, wouldn't any way of identifying 
the tangent space with it's dual be enough?

> It is crucial to note that some
> *extra geometrical structure* (the connection) is involved in the
> covariant derivative that is not needed for the Lie derivative.  
> 
> To repeat: The connection allows you to compare tangent vectors at two
> points of the manifold given only a curve connecting them; the Lie
> derivative requires a whole vector field, to generate a flow that "drags
> tensors along". 

Just as a bit of clarification:
The problem with taking derivatives in general is that you want to compare
the values of an object at two nearby points to see how it changes. In order
to do this, you must move the value of the object at place one to place two 
to compare it to the value of the object at place two. For vectors, and higher 
index objects, how you move the object will affect its value at place two 
and thus the result of your derivative. In flat space, one can simply use the 
formula that, for a vector in three space with cartesion components (x,y,z) 
say, if at point one = (x1,y1,z1) the vector field has value (x-x1,y-y1,z-z1)
then we move it to point two = (x2,y2,z2) by the prescription (x-x2,y-y2,z-z2).

In curved spaces, it is more difficult to obtain a prescription. Lawrence
has mentioned two. In the case of a Lie derivative, as John mentions, one
needs two vector fields. One is the vector field that we wish to take the
derivative of (say u). The other is the tangent field to the flow of curves 
generated by a one parameter group of diffeomorphisms (maps that preserve the 
manifold and it's tangent space) (say v). Roughly speaking, one uses the 
flow of integral curves of v to provide a path on which to transport
a copy of u at point 1 to point 2 for comparison. One does this by requiring 
that the inner product of tangent vector to the flow (v) and the vector of 
interest (u) not change; thus a unique prescription for comparing 
u at the two points that requires only a vector field v and a metric (so
that we can take inner products). However, it is important to remember that 
the two points must be connected by the flow lines. 

The second is covariant derivative. This time, rather than the tangent
field to a flow of curves generated by diffeomorphisms, we use a connection.
Given a connection, we can obtain the geodesics of a manifold. Two nearby
points will always have a unique geodesic connecting them, so we can now
use the geodesics and their tangents to (parallel) transport a copy of u
from point 1 in the same way we did with the Lie derivative. One big 
difference, however, is that point 2 can be in any direction now, rather than
just in the direction of a flow. Again, a unique way of taking a derivative,
only this time, since the direction is arbitrary, the result will be of one
higher rank than the u was to start with. One can then contract this new
index with any vector v defined at point 2 to find the component of this
derivative in the direction of v. In this case, it is important to remember
that you need only a vector v at one point, a connection, for the derivative,
and a metric to take inner products. Also, remember that there is an arbitrary
choice of connection, however the Levi-Civita connection (D_a g_bc = 0) is the
one most often used, in which case v and the metric will be enough.

So, Lawrence, the similarity is that they both use parallel transport to 
compare vectors at two points. The difference is in what you need to use
to transport them. 
--
########################## _`|'_ ##############################################
##  Warren G. Anderson     |o o| "... for its truth does not matter, and is  ##
##  Dept. of Physics       ( ^ )  unimaginable." -J. Ashbery, The New Spirit ##
##  University of Alberta  /\-/\      (anderson@fermi.phys.ualberta.ca)      ##