Parallel transport Definition and 71 Threads

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. cianfa72

    I Notion of congruent curve along a vector field

    Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##. Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the...
  2. 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...
  3. A

    I Parallel Transport of a Tensor: Understand Equation

    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...
  4. J

    I Parallel transport general relativity

    Suppose you have a tensor quantity called "B" referenced in a certain locally inertial frame (with four Minkowski components for instance). As far as I know, a parallel transportation of this quantity from a certain point "p" to another point "q" consists in expressing it in terms of the...
  5. LCSphysicist

    Finding ds² on a Cone: How to Use Geodesic Equations for Parallel Transport

    I am having too much trouble to solve this exercise, see: Using (R,phi,z) ub is the path derivative U is the path V is the vector $$V^{a};_{b}u^{b} = (\partial_{b}V^{a} + \Gamma^{a}_{\mu b} V^{\mu})u^{b}$$ $$U = (0,\theta,Z)$$ I am not sure what line element to use, i mean, a circle around a...
  6. 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?
  7. A

    A Parallel transport on a symplectic space

    Sorry if the question is not rigorously stated.Statement: Let ##(q,p)## be a set of local coordinates in 2-dimensional symplectic space. Let ##\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})## be a set of local coordinates of certain open set of a differentiable manifold ##\mathcal{M}.## For...
  8. S

    B Exploring Differential of Vector Component vs Change During Parallel Transport

    I'm reading 'Core Principles of Special and General Relativity' by Luscombe - the part on parallel transport. I guess ##U^{\beta}## and ##v## are vector fields instead of vectors as claimed in the quote. Till here I can understand, but then it's written: I want to clarify my understanding of...
  9. 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...
  10. 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...
  11. 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...
  12. R

    I Parallel Transport & Geodesics: Explained

    I am currently reading Foster and Nightingale and when it comes to the concept of parallel transport, the authors don't go very deep in explaining it except just stating that if a vector is subject to parallel transport along a parameterized curve, there is no change in its length or direction...
  13. P

    A Parallel transport of a 1-form aound a closed loop

    Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...
  14. B

    I Parallel transport of a vector on a sphere

    question1 : if you draw a small circle around the north pole (it should be the same at every points because of the symmetry of the sphere),then it is approximately a flat space ,then we can translate the vector on sphere just like what we have done in flat space(which translate the vector...
  15. J

    I About the solution of the parallel transport equation

    If a vector moves along a particular curve ##l## from point ##x_0## to point ##x## on a manifold whose connection is ##\Gamma^i_{jk}(x)##, then the vector field we get obviously satisfy the pareallel transport equations: $$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=0$$ Because ##[\Gamma^i_{jk}(x)...
  16. K

    I Parallel Transport: Uses & Benefits

    What is the usefulness of parallel transporting a vector? Of course, you can use it to determine whether a curve is a geodesic, but aside from that, what can it be used for?
  17. A

    I Parallel transport of tangent vector....(geodesic)

    https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures_2017.pdf&ved=2ahUKEwi468HjtNbgAhWEeisKHRj9DNEQFjAEegQIARAB&usg=AOvVaw3UvOQyTwkcG7c7yKkYbjSp&cshid=1551081845109 Here in page 55 it is written that geodesic is a curve whose tangent...
  18. J

    A Berry phase and parallel transport

    Hello. In the following(p.2): https://michaelberryphysics.files.wordpress.com/2013/07/berry187.pdf 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...
  19. 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...
  20. J

    A Global solution to parallel transport equation?

    In general relativity, a vector parallel along a curve on a manifold M with a connection field Γ can be expressed: ∂v+Γv=0 We know that if the curvature corresponding to Γ is non-zero, which means if we parallel transport a vector along different paths between two points, we will get different...
  21. S

    I Lie Derivatives vs Parallel Transport

    Hello! In my GR class we were introduced to the parallel transport as the way in which 2 tensors can be compared with each other at different points (and how one reaches the curvature tensor from here). I was wondering why can't one use Lie derivatives, instead of parallel transport. As far as I...
  22. 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...
  23. davidge

    I Parallel transport vs Fermi Transport

    Since for a general contravariant vector, ##\nabla_{\nu}V^{\mu}## will not in general be zero, is it correct to say that all of them are transported by Fermi Transport? (With the only vector being parallel transported being the four velocity vector?)
  24. 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...
  25. P

    A More on Parallel Transport: Existence & Uniqueness

    The recent thread on parallel transport has raised a couple of things I'd like to review for my own sake. I'll address them one at a time as my time permits. The first question is this. If we offer ##t^a \nabla_a u^b## or the equivalent ##\nabla_{\vec{t}} u^b## as the definition of parallel...
  26. F

    I Conservation of dot product with parallel transport

    Hello, I have 2 questions regarding similar issues : 1*) Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle between the tangent vector and transported vector is...
  27. J

    I Connection between Foucault pendulum and parallel transport

    Hello! I try to think about the Foucault pendulum with the concept of parallel transport(if we think of Earth as being a perfect sphere) but I can't quite figure out what the vector that gets parallel transported represents(for example, is it the normal to the plane of oscillation vector?). In...
  28. mertcan

    I Taylor expansion and parallel transport

    hi, first of all in this image there is a fact that we have parallel transported vector, and covariant derivative is zero along the "pr"path as you can see at the top of the image. I consider that p, and r is a point and in the GREEN box we try to make a taylor expansion of the contravariant...
  29. snoopies622

    Calculate Parallel Transport: Get Out of Logical Loop

    I'm in a logical loop here: 1. A tensor undergoes parallel transport if, as it moves through a manifold, its covariant derivative is zero. 2. Covariant derivative describes how a tensor changes as it moves through a manifold. 3. A tensor undergoes change as it moves if it does not parallel...
  30. bcrowell

    Parallel transport to explain motion of light near black hole?

    I'm currently teaching a gen ed course called Relativity for Poets. This is the first semester I've taught it, and it's been a ton of fun so far. If anyone is curious, http://www.lightandmatter.com/area3phys120.html is the class's web page with links to the syllabus and lecture notes. The...
  31. D

    Parallel Transport & Covariant Derivative: Overview

    I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...
  32. E

    Confusion about parallel transport

    I am studying parallel transport in order to understand Berry curvature, but I know this topic is most commonly used in GR so I'm posting my question here. I do not know differential geometry. I am looking for a general explanation of what it means to parallel transport a vector. Mostly I am...
  33. JonnyMaddox

    Parallel transport on a cardioid

    Hi guys, I want to calculate an explicit example of a vector parallel transported along a cardioid to see what happens. Maybe someone could help me with that since no author of any book or pdf on the topic is capable of showing a single numerical example. So we need a vector field on a manifold...
  34. S

    Questions on Parallel Transport: Riemann Tensor & More

    In my recent studies of curvature, I worked with the Riemann tensor and the equation: (\deltaV)a= A\muB\nuRab\mu\nuVb Now previously, I worked in 2D with the 2 sphere. While doing so, I learned that if I set my x1 coordinate to be θ and my x2 coordinate to be ø, then the vectors that serve...
  35. D

    Parallel transport around a loop

    From the Wikipedia article on the Riemann curvature tensor: The last sentence assumes that ##\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}## returns Z back to x0. It isn't obvious to me that this should be true. My guess is that this is true because X and Y commute, but I can't think of a...
  36. P

    Equation for parallel transport involving sectional curvature

    This is an expression I came across in a paper I am going through. It involves an expression for the parallel transport of a tangent vector taking into consideration the sectional curvature of simply connected space-forms in \mathbb R^4 . I have not been able to derive it.The equation simply...
  37. L

    Geometric phase of a parallel transport over the surface of a sphere

    I have this question on the calculation of the geometric phase (Berry phase) of a parallel transporting vector over the surface of a sphere, illustrated by Prof. Berry for example in the attached file starting on page 2. The vector performing parallel transport is defined as ψ=(e+ie')/√2...
  38. pellman

    How do we parallel transport a vector?

    Given a curve c(τ) with tangent vector V, a vector field X is parallel transported along c if \nabla_V X=0 at each point along c. Let x^\mu(\tau) denote the coordinates of the curve c. In components the parallel transport condition is \frac{dx^\mu}{d\tau}\left(\partial_\mu X^\alpha +...
  39. P

    Does torsion make parallel transport direction dependent?

    Torsion has propped up in a couple of recent threads, but none of my texts really cover it well. Does torsion make parallel transport direction dependent? I.e. if we parallel transport some vector v "forwards" along a curve, and then "backwards" along the very same curve to its starting...
  40. Q

    Horizontal lift, or parallel transport

    Hello, everyone! I'm studying Nakahara's book, Geometry, Topology and Physics and now studying the connection theory. I come across a problem. Please look at the two attachments. In the attachment , Nakahara said we could use the similar method in the attachment to get \tilde X, but why...
  41. F

    How to understand Parallel Transport

    I'm listening to Prof. Leonard Susskind's lectures on GR on youtube.com at http://www.youtube.com/watch?v=hbmf0bB38h0&feature=relmfu He's trying to explain how to visualize parallel transport of a vector. But I'm having a hard time of it. I think I understand it. Let me know if I got it...
  42. L

    Horizontal Lift vs Parallel Transport in Principal Bundle & Riemannian Geometry

    I am a physicist trying to understand the notion of holonomy in principal bundles. I am reading about the horizontal lift of a curve in the base manifold of a principal bundle (or just fiber bundle) to the total space and would like to relate it to the "classic" parallel transport one comes...
  43. K

    Parallel transport and geodecics

    So from what I understand if you pass a vector (using parallel transport) through a closed curve where there is curvature in the interior, the vector will come back not to it's original vector but with a changed sense. However if the vector is on a geodesic it will not change its sense after it...
  44. R

    Coordinate-independence of equation for the parallel transport

    Homework Statement Please show that the defining equation for the parallel transport of a contravariant vector along a curve \dot{\lambda}^a+\Gamma^a_{bc}\lambda^b\dot{x}^c=0 is coordinate-independent, given that the transformation formula for the christoffel symbol being...
  45. Matterwave

    Parallel Transport: Does Quadrilateral Close Without Torsion?

    Assuming that we are working with an infinitessimally small region of a manifold so that we can consider only first order effects, does parallel transport in the absence of torsion necessarily "close the quadrilateral"? What I mean is, if I have two vectors (very small vectors) V and U, and I...
  46. M

    Parallel Transport on a sphere

    Homework Statement See attached picture of the problem. Homework Equations See attached picture of the pair of diff equations. The Attempt at a Solution I was able to solve the problem up to the point the picture gets to. However, the author says he obtained this by integrating...
  47. D

    Intuition for Parallel Transport

    Hi All, I am an idiot but I don't understand the parallel transport: Parallel transport should be in surface right? How to keep parallel and still be in surface?
  48. T

    Intuitive explanation of parallel transport and geodesics

    Hello, First of all, please excuse me if I posted in the inappropriate place.. While a student few years ago, I used to work a lot with advanced differential geometry concepts, but never got an intuitive view of HOW humanity got to think about parallel transport, why it contained two words...
  49. P

    Lie Derivatives and Parallel Transport

    Hi, I've begun learning about General Relativity, though I've already had some exposure to differential geometry. In particular, I understand that Lie Differentiation is a more "primitive" process than Covariant Differentiation (in that the latter requires some sort of connection). My...
  50. R

    Is Walking at Constant Latitude with Your Arm Pointing North Parallel Transport?

    If you walk at constant latitude with your arm always sticking towards the North pole, is that parallel transport of your arm? The equations don't seem to say it is. The vector field would be \vec{V}=V(\theta)e_\theta . The component of the vector only depends on theta, but at constant...
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