🚀 go-pugleaf

I've been struggling with the concept of covariant derivative for some time.
I'd like to add my $0.02 and my $0.50 questions to this thread.

Here goes,

A covariant derivative need only be an operation that adds one covariant
index to any tensor. Consider a Reimmanian space.

For a scalar I this is simply

          i
I  = dI/dx  , where the derivatives are partials.
 ;i

For a covariant tensor of 1st rank (order, whatever) A

            j   k
A   = dA /dx - G  A
 i;j    i       ij k

as I'm sure you all know. G can be formed from any non-singular covariant
rank 2 tensor. The "connection" is an additional restraint that need not
be imposed to insure the tensor character of the resulting tensor. 

Ok that was my $0.02 worth, now for the $0.50 questions,

John Baez writes:

>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.

Hmm.... If I look at the covariant derivative of the scalar I above, all
I need to know are the partial derivatives of I at the points that I want
the derivative. Let me just talk about one point. Then I need to know
I(x1,x2,,,) only, that is know the scalar as a function of the coordinates
at the point is question.

In the general case, I need to know G, which means I need to know the
tensor in question as a function of coordinates, and the covariant tensor
of rank 2 used for the derivative as a function of the coordinates, both
at the point in question.

One question is this, how does knowing a tangent vector at one point
give me enough information to compute a covariant derivative?

Another question is about the connection. As I stated above, I don't
see the need for a connection to establish a covariant derivative.
Is this just a matter of definition?

Enough for now.

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Ric Peregrino   c/o Hewlett Packard Co.    I represent only myself
ric@spd.hp.com  1501 Page Mill Rd Bldg 5M  I may be wrong, maybe not
415-857-7526    Palo Alto, CA 94304        Do DC photons exist?
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