🚀 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". 

I had replied:
: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?

Sorry, I just want to head off criticism at the pass here. Of course if you
define your connection using a vector field, that vector field lives in the
tangent space to the fiber bundle (in some cases T(TM)) rather than TM.  Of
course that's how you get the generalized "family of flows" - you have a
family of vectors corresponding to each point of TM, rather than just a
single vector field on TM.