Talk:Parallel transport

message: There are various levels of generality at which one can discuss parallel transport:
* for vector fields;
* for sections of arbitrary vector bundles;
* for sections of arbitrary geometric bundles (e.g. frame bundles and their associated bundles);
* for sections of arbitrary fiber bundles (e.g. principal bundles and their associated bundles);
At what level should this article employ most of its energies? Geometry guy 16:08, 12 March 2007 (UTC)
replies: text: Why not all of them?  But if I had to pick one, I believe it should focus primarily on vector fields since that is likely to be the area of interest to most people looking in the general community.  At least, it should start out with vector fields.  The other levels of generality are also important, though, for the article holonomy (which also needs a lot of work). Silly rabbit 14:11, 13 April 2007 (UTC)
thread_title: Generality of this article?

message: I'm trying to understand how in general one transports tensors in the tangent space of a point in a Lie group to some other point in a Lie group, particularly using matrix representation. For example, the exponential map on SO(3) lets you represent tangent vectors in SO(3) as skew-symmetric matrices, and there is a 1:1 mapping between 3x3 skew-symmetric matrices and 3-vectors, however they transform differently... If I have an axis-angle vector, \omega, I can rotate it (transport it) using a rotation matrix: R\omega. However, if I have the skew-symmetric representation \omega, [\omega]_\times, I believe it transforms like a matrix; that is,
:[R\omega]_\times = R[\omega]_\times R^\top.
Two questions:
# In SO(3), it seems like the "matrix" representation of a "second-order" tensor would be a (3x3)x(3x3) matrix. How would that transform? Presumably you'd need to transform both 3x3 portions, so it would look something like R_{ij}R_{nk}T_{jk\ell m}R_{\ell p}R_{qm}, which seems pretty kludgy.
# Outside of SO(3), the exponential map on a matrix gives you tangent "vectors" that are matrices with no obvious "simpler" representation like an axis angle. There must be a wholistic way of transporting tangent vectors and tensors. Is it simpler than constructing some high-order object and transforming along each index as above? —Ben FrantzDale (talk) 02:42, 5 November 2009 (UTC)
thread_title:  Transporting tensors on Lie groups 

message: It would be useful to include the equation for parallel transport of a vector as used in general relativity:
\frac{dv^\mu}{ds} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{ds}v^\beta = 0
But where to put it in the article? 193.198.162.14 (talk) 09:49, 6 February 2018 (UTC)
thread_title:  Equation for parallel transport in general relativity 

message: It appears that the vectors illustrating the parallel transport are viewed form a point above the equatorial plane.  The vector at B is shown properly.  The direction of the vector between A and B is not correct.  It should be tangent to the equator, i.e. showing downward from the equator, just like the vector at B.  Also, the vector at A showing to the N-S line is in error.  It should be tangent to the equator, i.e. directed downward from the equator.  It is absolutely misleading to end it at that line.  It has nothing to do to the N-S line.178.48.163.158 (talk) 10:56, 20 August 2020 (UTC)
thread_title:  Illustration of Parallel Transport