parallel transport
On a flat table you can slide an arrow from one place to another while keeping it pointing the same way — moving it parallel to itself is unambiguous. On a curved surface this is suddenly subtle: there is no global notion of 'the same direction' at two distant points, because the surface tilts in between. Parallel transport is the careful rule for carrying a vector along a chosen path while turning it as little as the curved geometry allows — keeping it as parallel to itself as the surface permits, step by step.
The precise definition uses the covariant derivative: a vector V is parallel transported along a curve when its covariant derivative along the curve's tangent vanishes, nabla_(tangent) V = 0 at every point. In components this is a system of ordinary differential equations, dV^k/dt + Gamma^k_{ij} (dx^i/dt) V^j = 0, that you integrate along the path; the Christoffel terms continuously rotate the components just enough to compensate for the turning basis, so the vector never makes a 'real' turn of its own. Parallel transport preserves the vector's length and the angle between any two vectors carried together (a consequence of metric compatibility). The startling fact is that the result is path-dependent: carry a vector around a closed loop on a curved surface and it generally comes back rotated, even though at every step you turned it as little as possible. The angle of that mismatch is a direct, hands-on measure of the curvature enclosed by the loop.
Parallel transport is the operational heart of curvature and the bridge to the Riemann tensor: the rotation a vector suffers around an infinitesimal loop, divided by the loop's area, IS the Riemann curvature acting on the vector. It explains the precession of a gyroscope orbiting the Earth (measured by Gravity Probe B), the rotation of the Foucault pendulum's swing plane as the Earth turns (a parallel-transport holonomy on a sphere), and Berry's geometric phase in quantum mechanics. The honest, memorable takeaway: on a curved space there is no path-independent way to compare directions at separated points — the answer depends on the route you took, and that path-dependence is curvature made tangible.
Start at the North Pole holding an arrow, walk down to the equator keeping it parallel, walk a quarter way around the equator, then walk back up to the pole — always parallel-transporting. The arrow returns pointing 90 degrees away from how it started, even though you never deliberately turned it. That 90-degree deficit equals the enclosed area times the sphere's Gaussian curvature: the sphere's curvature, read off a walk.
A vector carried around a spherical triangle returns rotated — the rotation angle is the enclosed curvature.
Parallel transport on a curved space is path-dependent: the same vector carried to the same endpoint along two different routes generally arrives pointing different ways. This is not an error in the procedure — the discrepancy is precisely the signature of curvature, and it vanishes only in flat space.