Talk:Connection (vector bundle)

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This article enriches the connections category but needs refinement. Geometry guy 22:44, 29 March 2007 (UTC)

[edit] Material from covariant derivative not yet incorporated

One advantage of defining the connection in this way is that the following theorem becomes a statement about derivations on Lie algebras, to which one may then apply purely algebraic techniques from Lie algebra cohomology:

There exists a local trivialization of the bundle E with a basis of parallel sections if, and only if, the curvature vanishes identically.

If P(E) → M is the frame bundle, consisting of all bases of E under the action of G = GL(r) where r is the rank of E, then an Ehresmann connection on P(E) induces a Koszul connection on E as follows. There is a natural one-to-one correspondence between (local) sections of E and functions φ : P(E) → R^r (defined locally over the base) which are equivariant under the action of G. Let the function φ associated to V be denoted f(V) for each local section V of E. The associated Koszul connection may be defined by

$f(\nabla_X V) = L_{X^{Hor}}[f(V)]$

where X^Hor is the horizontal lift of the vector field X, and L is the Lie derivative in the total space of P(E). (Included here. Silly rabbit 18:30, 31 May 2007 (UTC))

If a connection in E (or a principal bundle associated with E) is specified by means of a parallel translation along curves, then a Koszul connection can be identified with the derivative of parallel translation. Let x_t be a curve in M, and let

$\tau_0^t:E_{x_t}\rightarrow E_{x_0}$

be the parallel translation in the fibres. Then

$\nabla_{\dot{x}_0} V = \lim_{t\rightarrow 0} \frac{\tau_0^t(V_{x_t})-V_{x_0}}{t}$

defines the associated Koszul connection. (Included in parallel transport article. Silly rabbit 18:30, 31 May 2007 (UTC))

(This material needs some work before it is suitable for inclusion here.) Geometry guy 10:58, 16 February 2007 (UTC)

Actually, your second point above is already included. See the properties section of the article. -- Fropuff 17:44, 16 February 2007 (UTC)

Very good. I just cut and paste this stuff in to make sure it didn't get lost (accidently - it may not be worth including) in the new structure. Anyway, I think it would be nice to draw out the relation with principal connections (in both directions) more visibly at some point. Geometry guy 00:55, 18 February 2007 (UTC)

Cf also the following from connection form.

The connection form for the vector bundle is the form on the total space of the associated principal bundle, but it can also be completely described by the following form (on the base in a not invariant way). This subsection can be considered as a smoother but somewhat inaccurate introduction to connection forms...

If one chooses a local trivialization of the vector bundle and takes $\nabla'$ to be the corresponding trivial connection, then $ω$ gives a complete local description of $\nabla$ .

The choice of trivialization is equivalent to choosing frames in each fiber; this explains the reason for the name method of moving frames. Let us choose (a local smooth section of) basis frames $e i$ in fibers. Then the matrix of 1-forms $\omega=\omega_i^j$ is defined by the following identity:

$\nabla_u e_i=\sum_j\omega^j_i(u)e_j.$

If $G\subset GL(F)$ is the structure group of the vector bundle and the connection $\nabla$ respects the group structure then the form

ω

is a 1-form with values in

g

, the Lie algebra of

G

Geometry guy 11:06, 16 February 2007 (UTC)

[edit] curvature - change in parallel transport

As Connection (vector bundle) seems to cover curvature, too, I'll ask here.

Is there a formula like:

Assume there is a smooth family of smooth curves $γ t (s)$ with $γ t (0) = p$ and $γ t (1) = q$ for some p and some q, points in a manifold M. Let E be a vector bundle over M with a connection. Let $P_{t,s_1,s_2}:E_{\gamma_t(s_1)}\to E_{\gamma_t(s_2)}$ be the parallel transport along $γ t$ for $s 1$ to $s 2$ . Then one has for some $V\in E_p$ $\frac{d}{dt}P_{t,0,1} |_{t=0} V = \int_{0}^1 P_{t,s,1} |_{t=0} R(\frac{d}{ds'}\gamma _t(s')|_{s'=s},\frac{d}{dt'}\gamma _{t'}(s)|_{t'=T})P_{t,1,t} |_{t=0}Vds$ I can't find it, but it seems interesting to me, in order to understand the connection between curvature and parallel transport along different curves. One could then integrate over t or leave the V or take p=q.

Am I missing out on something? Is this in every text book and I'm just overlooking it. Or is it not in every textbook, because it's maybe both wrong and irrelevant? Thanks, JanCK 17:28, 1 March 2007 (UTC)

You may be able to get something (in case you don't already know) from the physics approach to parallel transport via Wilson lines and path-ordered exponentials. Geometry guy 21:51, 20 March 2007 (UTC)