parallel transport Definition and Topics - 10 Discussions

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.
The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, vice versa, parallel transport is the local realization of a connection.
As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose–Singer theorem makes explicit this relationship between curvature and holonomy.
Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.

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  1. George Keeling

    I How do a bunch of integrals make an n-simplex or an n-cube?

    This question arises from Carroll's Appendix I on the parallel propagator where he shows that, in matrix notation, it is given...
  2. A

    I Parallel transport of a tensor

    According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is: ##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0## Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
  3. steve1763

    I Parallel transport on flat space

    When parallel transporting a vector along a straight line on flat space, does the connection (when calculating the covariant derivative) always equal zero? Do things change at all when using an arbitrary connection, rather than Christoffel symbols?
  4. cianfa72

    I Parallel transport vs Lie dragging along a Killing vector field

    Hi, I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario. Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist...
  5. cianfa72

    I About Covariant Derivative as a tensor

    Hi, I've been watching lectures from XylyXylyX on YouTube. I believe they are really great ! One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
  6. M

    Solving the same question two ways: Parallel transport vs. the Lie derivative

    a) I found this part to be quite straight forward. From the Parallel transport equation we obtain the differential equations for the different components of ##X^\mu##: $$ \begin{align*} \frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\ \frac{\partial...
  7. J

    A Berry phase and parallel transport

    Hello. In the following(p.2): Berry uses parallel transport on a sphere to showcase the (an)holonomy angle of a vector when it is parallel transported over a closed loop on the sphere. A clearer illustration of this can be...
  8. J

    A Is the Berry connection a Levi-Civita connection?

    Hello! I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric). When performing a parallel transport with the L-C connection, angles and lengths are...
  9. J

    A Can you give an example of a non-Levi Civita connection?

    Hello!! Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection. So, I wanted to know of any examples of...
  10. A

    I Understanding Parallel Transport

    I'm currently in a GR class and have come across the notion of parallel transport, and I've searched and searched the last few days to try and understand it but I just can't seem to wrap my head around it, so I'm hoping someone here can clarify for me. The way I picture parallel transport is...