parallel transport

Take an arrow and slide it along a path, at every step keeping it pointing 'as parallel as it can' to how it pointed a moment before, without ever deliberately twisting it. On a flat table this is trivial: the arrow returns to its start unchanged. On a curved surface something startling happens. Walk an arrow from the equator up to the North Pole, across to a different longitude, and back down to the equator, and it comes home rotated, even though you never turned it. That rotation is your direct, hands-on encounter with curvature.

Precisely, a vector V is parallel-transported along a curve with tangent u when its covariant derivative along the curve vanishes: u^a nabla_a V^b = 0, equivalently dV^b/dlambda + Gamma^b_ac (dx^a/dlambda) V^c = 0. This is the natural generalization of 'keeping a vector constant' to a space where constancy in the naive sense is not coordinate-independent. In a curved space the result of transport depends on the path taken; carry the same vector between two points by two different routes and you generally get two different answers.

Parallel transport gives curvature an operational meaning: the net rotation a vector suffers on being carried around a closed loop is proportional to the curvature enclosed and to the loop's area, a relationship named holonomy. It also gives geodesics a clean definition: a geodesic is the one curve that parallel-transports its own tangent vector, a path that always heads 'straight ahead' by its own reckoning. The caveat: only in flat space is transport path-independent, so a rotation after a round trip is not error, it is the signal you are looking for.