Parallel Transport

Prerequisites: Manifold. Full path (3 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

see also connections

Intuitive

To understand parallel transport imagine you're holding a long stick in your hand. You are then asked to walk a given path while keeping the stick straight all the time. What you are doing then is a parallel transport of the stick.

The important thing is now that if you do this and walk a closed curve, i.e. you end up where you started, the final position of the stick will not be the same as the initial position. This happens no matter how good you are in keeping the stick straight.

For example, consider two different paths around a ball as shown in the image below.

If you follow the path on the left-hand side, where the position of the stick corresponds to the yellow arrows, the final position of the stick is not the initial position, since a ball is curved and not flat.

Parallel transport on a sphere can best be understood by imagining the sphere to be rolling on a flat surface. Suppose there are straight lines and curved lines drawn on the flat surface in wet ink. And suppose there are arrows spaced frequently along the line, all pointing, say, to the lower left. The term "parallel transport" is based on this picture. Since the arrows in the plane are all parallel, the arrows printed onto the sphere exemplify parallel transport along the printed curve on the sphere.

Given a curve on the sphere and an arrow representing a direction at the start of the ruve, one can place the sphere on the plane so that the starting point is the point of tangency. Imagine there is an arrow at each point in the plane drawn in ink and parallel to the original arrow. If the sphere is rolled along the plane so that it prints the curve on the sphere onto the plane, the arrows from the plane will also be printed onto the sphere. The latter arrows give a path in the bundle of directions that starts with the initial arrow and lies above the given curve.

When a straight line is printed onto the rolling sphere (or onto any other curved surface), the curve that it forms on the surface is called a geodesic.

Parallel transport can be described for a curved path made up of geodesic segments without reference to rolling. It can be carried out by maintaining a constant angle between the transported arrows and the tangents along successive geodesics. From a perspective above the surface of the sphere, however, parallel transport along a geodesic may seem anything but parallel. The arrows may appear to rotate. When an arbitrary curved line is printed onto the sphere, the rotation of the arrows as viewed from above the surface may appear to be even more chaotic.

Parallel transport provides a way of making quantitative and explicit the intuitive difference between a curved surface and a flat one. When an arrow is carried by parallel transport around a closed path in the plane, the directions of the arrow at the start and at the finish coincide. Parallel transport around a closed path on a curved surface, however, may not lead to such coincidence. If there is a change in the direction of the arrow when it completes a single circuit of a closed path, the angle between the final direction and the initial one is called the angular excess of the path. It follows from the way parallel transport is defined that the angular excess does not depend on the initial direction of the arrow.

Fiber Bundles and Quantum Theory by Bernstein and Phillips

Concrete

Let $\gamma(t)$, $t\in[0,1]$, be a curve on a manifold $M$ with coordinates $x^\mu$, and let $V^\mu(t)$ be a vector attached to each point $\gamma(t)$. Given a connection with Christoffel symbols $\Gamma^\mu_{\nu\rho}$, the covariant derivative of $V$ along the curve is

$$\frac{DV^\mu}{dt} = \frac{dV^\mu}{dt} + \Gamma^\mu_{\nu\rho}\,\frac{d\gamma^\nu}{dt}\,V^\rho .$$

$V$ is parallel transported along $\gamma$ if $DV^\mu/dt = 0$. This is a system of linear first order ODEs, so for every initial vector $V(0)\in T_{\gamma(0)}M$ there is exactly one solution. The map $V(0)\mapsto V(1)$ is a linear isomorphism $P_\gamma: T_{\gamma(0)}M \to T_{\gamma(1)}M$. In flat space with Cartesian coordinates $\Gamma=0$, the components stay constant and we recover the usual notion of "parallel". For the Levi-Civita connection of a metric $g_{\mu\nu}$, $P_\gamma$ preserves lengths and angles, and a geodesic is a curve that parallel transports its own tangent vector.

Example: a circle of latitude on the sphere. On a sphere of radius $R$ with $ds^2 = R^2(d\theta^2 + \sin^2\theta\, d\phi^2)$ the non-zero Christoffel symbols are $\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$ and $\Gamma^\phi_{\theta\phi}=\Gamma^\phi_{\phi\theta} = \cot\theta$. Along the circle $\theta=\theta_0$, $\phi = t$, the transport equations read

$$\frac{dV^\theta}{dt} - \sin\theta_0\cos\theta_0\, V^\phi = 0, \qquad \frac{dV^\phi}{dt} + \cot\theta_0\, V^\theta = 0 .$$

In terms of the components $a = V^\theta$ and $b = \sin\theta_0\, V^\phi$ with respect to the orthonormal directions $\hat e_\theta, \hat e_\phi$ this becomes $\dot a = \cos\theta_0\, b$, $\dot b = -\cos\theta_0\, a$. The vector rotates relative to $(\hat e_\theta, \hat e_\phi)$ with angular velocity $\cos\theta_0$. After one full turn, $t = 2\pi$, it has rotated by $-2\pi\cos\theta_0$, which equals $2\pi(1-\cos\theta_0)$ modulo $2\pi$. This is the area $2\pi R^2(1-\cos\theta_0)$ of the enclosed polar cap times the Gaussian curvature $1/R^2$. On the equator ($\theta_0 = \pi/2$), which is a geodesic, the vector comes back unchanged.

The example illustrates the general rule: for a small closed loop the change of a transported vector is, to leading order, the curvature (Riemann tensor) contracted with the vector and the area element spanned by the loop. A vanishing Riemann tensor means that transport around small loops does nothing.

Mathematically, to parallel transport a given vector we need a connection.

Abstract

Let $\pi: P \to M$ be a principal $G$-bundle with a connection, given as a $G$-equivariant horizontal distribution $H\subset TP$ with $T_pP = H_p \oplus V_p$, or equivalently as a connection one-form $\omega \in \Omega^1(P,\mathfrak{g})$ with $H = \ker\omega$. For a piecewise smooth curve $\gamma:[0,1]\to M$ and a point $p\in\pi^{-1}(\gamma(0))$ there is a unique horizontal lift $\tilde\gamma$ with $\tilde\gamma(0)=p$. The map $p\mapsto\tilde\gamma(1)$ is the parallel transport $P_\gamma: \pi^{-1}(\gamma(0))\to\pi^{-1}(\gamma(1))$. It commutes with the right $G$-action, is unchanged under orientation preserving reparametrisations of $\gamma$, and satisfies $P_{\gamma_2 * \gamma_1} = P_{\gamma_2}\circ P_{\gamma_1}$ and $P_{\gamma^{-1}} = P_\gamma^{-1}$. On an associated vector bundle $E = P\times_G V$ it induces linear isomorphisms between fibres, and the covariant derivative is recovered from it as

$$\nabla_{\dot\gamma(0)}s = \frac{d}{dt}\Big|_{t=0} P_{\gamma|_{[0,t]}}^{-1}\, s(\gamma(t)) .$$

In a local trivialisation with $\nabla = d + A$, $A$ a $\mathfrak{g}$-valued one-form acting in the representation $V$, a section $v(t)$ along $\gamma$ is parallel if $\dot v + A(\dot\gamma)\,v = 0$. The solution is the path-ordered exponential

$$v(1) = \mathcal{P}\exp\left(-\int_0^1 A(\dot\gamma(t))\,dt\right) v(0),$$

with later times standing to the left. Under a gauge transformation $v\mapsto gv$, $A\mapsto gAg^{-1} - (dg)g^{-1}$, the transport changes to $g(\gamma(1))\,P_\gamma\, g(\gamma(0))^{-1}$. For a closed loop based at $x$ the transport is conjugated by $g(x)$, so its trace in a representation, the Wilson loop, is gauge invariant. In physics conventions $A$ usually carries a factor $-i$ and the coupling constant, e.g. $A = -ig A^a_\mu T^a dx^\mu$ with Hermitian $T^a$.

Transport around loops based at $p$ defines the holonomy group $\mathrm{Hol}_p \subset G$; contractible loops give the restricted holonomy group $\mathrm{Hol}^0_p$, a connected Lie subgroup. By the Ambrose-Singer theorem the Lie algebra of $\mathrm{Hol}_p$ is spanned by the values $\Omega_q(X,Y)$ of the curvature two-form on horizontal vectors $X,Y\in H_q$ at all points $q$ that can be joined to $p$ by a horizontal curve. In particular $\mathrm{Hol}^0_p$ is trivial if and only if the connection is flat. For a flat connection $P_\gamma$ depends only on the homotopy class of $\gamma$ relative to its end points, and holonomy defines a homomorphism $\pi_1(M,x)\to G$ (up to conjugation). The Aharonov-Bohm phase is an example. For the Levi-Civita connection of an oriented Riemannian $n$-manifold, $\mathrm{Hol}\subset SO(n)$, and reductions to smaller groups correspond to parallel tensor fields (for example $U(n/2)$ for Kähler manifolds).

Why is it interesting?

The idea of parallel transport is important when we want to find out whether a given space is curved or not.

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold

Leads to: Connections, Geometric Phase, Wilson Loops.