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nLab > Latest Changes: parallel transport

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010

    wrote parallel transport (which was previously redirecting to connection on a bundle).

    Also rewrote Dyson formula.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 2nd 2010
    Great, a separate entry was needed for a while.
    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeSep 3rd 2010
    • (edited Sep 3rd 2010)

    Nice, but it would be even nicer if there was some slick nPOV statement right at the beginning like “Parallel transport is a functor…” or something.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2010
    • (edited Sep 3rd 2010)

    Okay, I added a section on the functorial description.

    This has now a bit of overlap with the discussion at connection on a bundle, but anyway.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2010

    added also a section on the meaning of parallel transport in physics

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeSep 8th 2010

    A related new entry geodesic.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    A related new entry geodesic.

    Thanks! I added a sentence briefly relating this back explicitly to parallel transport.

    • CommentRowNumber8.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 26th 2011
    I've posted my Parallel transport - revisited to the arXiv. Comments still welcome.
    N.B. This is NOT smooth parallel transport, though hopefully the exposition is reasonably so. :-)
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2020

    the References-section used to only point to a list of references on my personal web. I have now copied that over to here, and completed some of the publication data

    diff, v19, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2020

    added pointer to:

    • Piotr Hajac, Axiomatic holonomy maps and generalized Yang-Mills moduli space, Letters in Mathematical Physics volume 27, pages301–309 (1993) (doi:10.1007/BF00777377)

    diff, v19, current

  11. Corrected typo (“morphsims”)

    Frank

    diff, v26, current

  12. Corrected typo (“morphsims”)

    Frank

    diff, v26, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 26th 2025

    Added:

    Holonomic equivalence of curves

    \begin{definition} Two curves with the same endpoints are holonomically equivalent if for every principal G-bundle with connection, the holonomies along both curves coincide. \end{definition}

    This definition has many variants, depending on which curves are allowed (e.g., smooth, continuously differentiable, continuous, rough paths).

    For continuously differentiable paths we have the following characterization, see Theorem 2.11 in Bischoff–Lee for an exposition and further references.

    \begin{theorem} Suppose x is a continuously differentiable path [0,1]→M in a smooth manifold M with x′(0)=0, x′(1)=0. The following conditions are equivalent.

    • The holonomy of x for every principal G-bundle with connection with a semi-simple structure group G vanishes.

    • There is a C1 endpoint-preserving homotopy of rank at most 1 (a thin homotopy) that contracts x to a point.

    • There is a C1 endpoint-preserving homotopy h that contracts x to a point such that imh⊂imx.

    • The map x factors through an R-tree T, i.e., a metrizable space T such that for every embeddings a,b:[s,t]→T such that a(s)=b(s), a(t)=b(t) we have ima=imb.

    • There is a Lipschitz function h:[0,1]→[0,∞) (a height function) such that h(0)=h(1) and if h(s)=h(t)=infh|[s,t],thenx(s)=x(t)$.

    • The Chen path signature of x is trivial.

    • The path x is word-reduced in the sense of Tlas Theorem 1. \end{theorem}

    diff, v31, current

1 to 13 of 13

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