Connection Definition and 15 Discussions

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.

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  1. 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...
  2. Vyrkk

    A Covariant derivative and connection of a covector field

    I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...
  3. pairofstrings

    B What is the connection between x^2 and a square shape?

    Hello. The curve y = x2 is a parabola that looks like this: I have a shape Square that looks like this: What I am noticing is that if I consider the equation y = x2 and also the shape Square, I find that there is no connection between them but the equation y = x2 is pronounced as x-square...
  4. 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...
  5. 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...
  6. 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...
  7. MattRob

    Showing that Metric Connections transform as a Connection

    Homework Statement Show that the metric connection transforms like a connection Homework Equations The metric connection is Γ^{a}_{bc} = \frac{1}{2} g^{ad} ( ∂_{b} g_{dc} + ∂_{c} g_{db} - ∂_{d} g_{bc} ) And of course, in the context of Einstein's GR, we have a symmetric connection, Γ^{a}_{bc}...
  8. L

    A Question of Cartan's structure equation (in Principal bundle)

    I have question about the proof of Cartan's structure equation in the context of pincipal bundle in Nakahara's book. The attached image is taken from the book. To show that the curvature two-form ##\Omega ## satisfies ##\Omega (X,Y) = {d_P}\omega (X,Y) + [\omega (X),\omega (Y)]## The author...
  9. D

    A Confusion on notion of connection & covariant derivative

    I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative. As I understand it a connection ##\nabla...
  10. L

    Affine connection Γ in terms of tetrad

    Hi, some one know the expression of the affine connection Γ in terms of tetrad formalism? I would like also some references if it's possible, i found a hit but i think that is wrong... please help me it's so important!
  11. loops496

    Levi-Civita Connection and Derivative of The Ricci Scalar

    I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...
  12. Natanijel

    Audio Jack Mystery

    Hello everyone :) I came across a strange situation today when I was listening to my music with my headphones on. When I pressed play to a song, the instrumental intro started off fairly normally (although the audio sounded a little "thinner" than usual, as if compressed), and then, when...
  13. U

    Flat Space - Christoffel symbols and Ricci = 0?

    Homework Statement [/B] (a) Find christoffel symbols and ricci tensor (b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##. Homework Equations The Attempt at a Solution Part(a) [/B] I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} =...
  14. G

    Non-affine Connections

    "Everyone" knows what an affine connection on a smooth manifold is a.k.a. covariant derivative. My questions are: i) Why are those connections called affine? ii) Is there a mathematical object that 'connects neighboring tangent spaces', that could be termed a 'non-affine connection'?
  15. D

    Challenge Connecting Multiple Smartphone Mic's Simply & Quickly

    Dear Readers, I'm facing a pretty interesting challenge, which is: "How to connect multiple smartphone mic's in the quickest and simplest way?" Here is a use case. I go into a meeting with a group of random (strangers) that all have smartphones, Android, iOS, Windows. Now I want to record this...