🚀 go-pugleaf

In article <1drukoINN801@darkstar.UCSC.EDU> ask@ucscb.UCSC.EDU (Andrew Stanford Klingler) writes:

>JB:
>
>>
>>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". 
>>
>The distinction you're making sounds kind of phony; after all one can
>define a connection using a vector field or a field of one-forms (I may
>be limiting myself to principal fiber bundles here).  It seems like the
>flow is just buried in the structure for a connection (in some sense a
>connection gets you "more than a flow" - the local family of geodesics
>through a point).  Without a connection, you have no way to compare vectors
>at two different points, so you have to transport tensors out, then back
>to the same point and compare them there.  With a connection, you have
>"flow in every direction" (geodesics thru the point) so parallel transport
>vectors from a point to any other point and compare them there.  From this
>point of view Lie derivatives and covariant derivatives represent the
>same geometric thing, I think.  Am I way wrong here?
>
>

Jon Baez, I hope, will have a more complete explanation, but yes
you're way off here. The covariant derivative and the Lie derivative
are very different things.

For one thing, a connection is something that can live in any
vector or principal bundle, whereas the Lie derivative is
only well defined for bundles associated to the tangent bundle.

Secondly, the connection is an additional structure that one
imposes on a bundle; the Lie derivative depends only on
the smooth structure on the manifold.

The easiest thing to do is to ponder this: On a tangent bundle
with a torsion free connection (like the Levi-Civita connection
of a Riemannian metric), 
The Lie derivative of a vector field W with respect to a vector
field V is

L_V W = [V,W] = D_V W - D_W V,

where L denotes Lie derivative and D denotes covariant derivative.

Addressing your discussion a little more directly, the point is
that whereas a connection allows you to parallel translate a
single vector from one point to another, if you have a global
vector field, you can't transport vectors between two arbitrary
points, but you can transport it between two points that lie
on the same integral curve of the vector field. In other words,
you can push forward a second vector field using the 1-parameter
flow of diffeomorphisms induced by the first vector field.
The Lie derivative is defined using this means of transporting
vectors.