On a flat plane, you can move a vector from one point to another and its direction is unambiguous — just slide it. On a curved surface, “sliding” is meaningless. There’s no global notion of “same direction.” Parallel transport is the best available substitute: move the vector along a curve, keeping it as straight as possible at each infinitesimal step.

The rule

A vector $v^\mu$ is parallel-transported along a curve $x^\mu(\lambda)$ if:

$$\frac{Dv^\mu}{d\lambda} = \frac{dv^\mu}{d\lambda} + \Gamma^\mu{}_{\alpha\beta} v^\alpha \frac{dx^\beta}{d\lambda} = 0$$

At each step, the Christoffel symbols correct for the changing coordinate basis. The result: $v^\mu$ changes components to compensate for the curvature, keeping the “intrinsic direction” constant.

Why it reveals curvature

On a flat surface, parallel transport around any closed loop returns the vector to its original direction. On a curved surface, it doesn’t — the vector comes back rotated by an angle that depends on the enclosed curvature.

This is the definition of the Riemann tensor. Transport $v^\mu$ around an infinitesimal parallelogram with sides $\delta a^\alpha$ and $\delta b^\beta$:

$$\delta v^\mu = R^\mu{}_{\nu\alpha\beta} \, v^\nu \, \delta a^\alpha \, \delta b^\beta$$

If $R = 0$, the vector returns unchanged — the space is flat. If $R \neq 0$, the vector rotates — there’s curvature.

The classic demonstration

Transport a vector around a triangle on a sphere:

1. Start at the equator (0°N, 0°E), pointing east
2. Walk north to the pole — the vector still points “east” (now toward 90°E)
3. Walk south along the 90°E meridian to the equator
4. Walk west along the equator back to the start

The vector now points north — it’s rotated 90° from its starting direction. The triangle encloses 1/8 of the sphere’s surface, and the rotation is 90° = π/2 radians, which equals the solid angle subtended.

For a general triangle with area $A$ on a sphere of radius $R$: the rotation angle is $A/R^2$ — the angular excess.

Physical consequences

Geodesic deviation: Two freely falling particles start with parallel velocities. As they travel through curved spacetime, their separation changes — this is tidal stretching. The rate of change is governed by parallel transport of the separation vector along the geodesic.

Gravity Probe B (2004–2011): Measured the precession of gyroscopes in orbit around Earth. A spinning gyroscope parallel-transports its spin axis along its worldline. In curved spacetime, this axis precesses — the geodetic effect (6.6 arcsec/yr) and the frame-dragging effect (0.039 arcsec/yr). Both matched GR’s predictions.