Parallel Transport

Core Idea

Parallel transport moves a tangent vector along a curve while keeping it "as constant as possible" according to a connection — formally, the covariant derivative along the curve vanishes. On a flat manifold, parallel transport is path-independent: the transported vector depends only on the endpoints. On a curved manifold, parallel transport around a closed loop generally rotates the vector, and the amount of rotation is directly related to the curvature enclosed by the loop. This path-dependence of parallel transport is the geometric essence of curvature.

Explainer

In Euclidean space, moving a vector from one point to another "without changing it" is obvious — you translate it, keeping its components constant. On a curved manifold, there is no such natural translation. Different paths from p to q might "hold the vector constant" in different ways, depending on the curvature encountered. Parallel transport resolves this ambiguity given a connection: it provides a specific rule for moving vectors along a specific curve.

Given a connection ∇ and a curve γ : [a,b] → M, a vector field V along γ is parallel if ∇_{γ'(t)} V(t) = 0 for all t. In coordinates, this is the system of ODEs dVᵏ/dt + Γᵏᵢⱼ (dγⁱ/dt) Vʲ = 0. The theory of linear ODEs guarantees a unique solution for any initial condition V(a) = v₀ ∈ T_{γ(a)}M. The map P_{a→b} : T_{γ(a)}M → T_{γ(b)}M sending v₀ to V(b) is the parallel transport map along γ. It is a linear isomorphism between tangent spaces. When the connection is the Levi-Civita connection, parallel transport is an isometry — it preserves lengths and angles.

The holonomy of a connection is what happens when you parallel transport around a closed loop. Starting with a vector v at a point p, transporting it around a loop γ back to p, you generally get a different vector P_γ(v) ≠ v. The set of all such rotations forms the holonomy group Hol_p(∇) ⊂ GL(TpM). On a Riemannian manifold, holonomy lies in O(n) (the orthogonal group) because parallel transport preserves the inner product. The holonomy group encodes global geometric information: for instance, Berger's classification of holonomy groups constrains the possible geometries of irreducible Riemannian manifolds.

The connection between parallel transport and curvature is the deepest insight. For an infinitesimal parallelogram spanned by vectors X and Y at a point p, transporting a vector v around the parallelogram produces a rotation approximately equal to R(X,Y)v, where R is the curvature tensor. Curvature measures the infinitesimal holonomy — the rotation per unit area of parallel transport around small loops. Zero curvature means parallel transport is locally path-independent: the vector you get at the end depends only on the endpoints, not on the path. Nonzero curvature means the manifold is genuinely curved, and the geometry felt by a vector depends on the path it takes.