Parallel Transport — Mathematics Knowledge System

Overview

Parallel transport is the process of moving a tangent vector along a curve on a manifold in such a way that the vector remains 'constant' with respect to the connection — its covariant derivative along the curve is zero. On a curved manifold, parallel transport is path-dependent: transporting a vector around a closed loop generally returns a different vector, and this failure of path-independence is measured by the curvature of the connection.

Intuition

Imagine carrying an arrow along the surface of a sphere while keeping it 'as parallel as possible' to itself — never turning it relative to the surface. If you carry it from the north pole to the equator, then a quarter way around the equator, then back to the north pole, the arrow comes back rotated. On flat space the arrow would return unchanged. The angle of rotation measures how curved the surface is — this is holonomy, and it is encoded by the Riemann curvature tensor.

Formal Definition

Definition

Let (M, ∇) be a smooth manifold with a connection ∇, and let γ: [a,b] → M be a smooth curve. A vector field V along γ is parallel if its covariant derivative along γ vanishes.

\frac{DV}{dt} = \nabla_{\dot{\gamma}(t)} V = 0 \quad \forall\, t \in [a,b]

V is parallel along γ if its covariant derivative along the velocity vector vanishes

Parallel transport equation

\frac{dV^k}{dt} + \Gamma^k_{ij}\, \dot{\gamma}^i\, V^j = 0

A system of linear ODEs for the components V^k(t)

In local coordinates (Christoffel symbols)

P_{\gamma,a}^{b} : T_{\gamma(a)}M \to T_{\gamma(b)}M

The linear isomorphism sending V₀ at γ(a) to the parallel translate V(b) at γ(b)

Parallel transport map

Theorems

Theorem 1: Existence and Uniqueness of Parallel Transport

\text{For any smooth curve } \gamma: [a,b] \to M \text{ and any initial vector } V_0 \in T_{\gamma(a)}M, \text{ there exists a unique parallel vector field } V \text{ along } \gamma \text{ with } V(a) = V_0.

Theorem 2: Parallel Transport is a Linear Isometry

\text{If } \nabla \text{ is the Levi-Civita connection of a Riemannian metric } g, \text{ then } P_{\gamma,a}^b \text{ is an isometry: } g_{\gamma(b)}(P V, P W) = g_{\gamma(a)}(V, W).

Worked Examples

1
Parameterize S² with spherical coordinates (θ, φ). The meridian at φ = 0 is γ(t) = (θ = t, φ = 0), with velocity γ̇ = ∂/∂θ.
$\dot{\gamma}(t) = \frac{\partial}{\partial \theta}$
2
The non-zero Christoffel symbols of S² in (θ,φ) coordinates are Γ^φ_{θφ} = Γ^φ_{φθ} = cot θ and Γ^θ_{φφ} = -sin θ cos θ.
$\Gamma^{\phi}_{\theta\phi} = \cot\theta, \quad \Gamma^{\theta}_{\phi\phi} = -\sin\theta\cos\theta$
3
Write V(t) = V^θ(t) ∂/∂θ + V^φ(t) ∂/∂φ. The parallel transport equations along γ (where γ̇^θ = 1, γ̇^φ = 0) are:
$\frac{dV^\theta}{dt} + \Gamma^\theta_{\theta\theta} V^\theta + \Gamma^\theta_{\phi\theta} V^\phi = 0$
4
Since Γ^θ_{θθ} = 0 and Γ^θ_{φθ} = 0, we get dV^θ/dt = 0, so V^θ is constant. Similarly dV^φ/dt = 0. Thus V = V₀ = ∂/∂θ is transported to itself: it remains ∂/∂θ at each point of the meridian.
$V(t) = \frac{\partial}{\partial \theta}$

✓ Answer

Along a meridian of S², the vector ∂/∂θ is parallel-transported to itself — it stays tangent to the meridian the whole way. No rotation occurs along a geodesic.

Practice Problems

Mediumfree response

On a flat manifold (ℝⁿ with Euclidean connection), describe the parallel transport of any vector along any curve. What does this say about holonomy?

Mediumproof writing

Prove that the parallel transport map P_{γ,a}^b preserves the inner product when ∇ is the Levi-Civita connection.

Quiz

Summary

Parallel transport moves a tangent vector along a curve by requiring its covariant derivative to vanish: DV/dt = 0.
In local coordinates this is a linear ODE system: dV^k/dt + Γ^k_{ij} γ̇^i V^j = 0, guaranteeing existence and uniqueness.
On a Riemannian manifold with the Levi-Civita connection, parallel transport is a linear isometry between tangent spaces.
Transporting around a closed loop generally returns a rotated vector — this holonomy measures curvature.
On S², a latitude-circle loop at angle θ₀ produces holonomy rotation 2π cos θ₀, equaling the enclosed solid angle.