🚀 go-pugleaf

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?