🚀 go-pugleaf

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.

It always gets me when folks ask math questions on sci.physics and *not*
sci.math.  Is this meant to indicate that the answers mathematicians
give will be too confusing?  Let's see - I'm a mathematician and I will
try not to be too confusing.  I won't repeat the definitions of
covariant vs Lie derivatives of tensor fields; instead, I'll just
emphasize some ways in which they are VERY DIFFERENT.  To begin with,
though, they are both defined for any sort of tensor field on a manifold
- so no "different type of topological space" is involved.

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.  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".