🚀 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?

I can't really estimate your rightness or wrongness.  Of course both
derivatives amount to ways of comparing vectors at different points.
However, the Lie derivative L_v w(x) at the point x of a vector field w
with respect to a vector field v does NOT just depend on the value of v
at the point x, while the covariant derivative D_v w(x) only depends on
the value of v at the point x.  The price one pays, of course, is that
to define D_v w(x) we need an extra structure, the connection.  But on
the up side, the covariant derivative is defined for all vector bundles
(equipped with connection) while the Lie derivative only works for the
tangent bundle and its relatives (tensor bundles).  

Another way of saying it, from a more global and algebraic perspective,
is that D_v w is linear in v (thinking of vector fields as a module over
the ring of smooth functions) while L_v w is not.