the Levi-Civita connection
/ LAY-vee chee-VEE-tah /
Here is a real problem on a curved space: how do you differentiate a vector field? To take a derivative you compare a vector at one point with the vector at a nearby point and divide by the gap — but those two vectors live in different tangent spaces, and there is no automatic way to compare them. A connection is the missing rule that says how to carry a vector from one tangent space to a neighbouring one so the comparison makes sense. The Levi-Civita connection is the one special, canonical connection that a Riemannian metric singles out.
Precisely, a connection provides a covariant derivative, written nabla_X Y, the rate of change of a vector field Y as you move in the direction of a vector field X, with the twisting of the tangent spaces properly accounted for. Many connections exist on a manifold, but the Levi-Civita connection is the unique one satisfying two natural demands: it is metric-compatible (parallel transport preserves lengths and angles, so the ruler does not change as you carry vectors around) and torsion-free (symmetric, meaning nabla_X Y - nabla_Y X equals the Lie bracket [X, Y], with no extra built-in spin). The Fundamental Theorem of Riemannian Geometry guarantees that exactly one connection meets both conditions, and an explicit formula (the Koszul formula) builds it from the metric alone; in coordinates its components are the Christoffel symbols.
This connection is the engine of the whole theory. Geodesics are the curves whose own velocity is parallel along them (nabla of velocity by velocity is zero); curvature is the failure of second covariant derivatives to commute; parallel transport, holonomy, and the equations of general relativity are all written through it. Its existence is what lets calculus on curved spaces be done canonically, with no arbitrary choices — the geometry of the metric dictates a single correct way to differentiate.
On the flat plane in ordinary x, y coordinates the Levi-Civita connection is trivial: nabla_X Y is just the componentwise ordinary derivative, because the coordinate axes never turn. Switch to polar coordinates and the connection is no longer trivial — the basis vectors d/dr and d/dtheta rotate as you move, so extra correction terms (the Christoffel symbols) appear, even though the underlying plane is still perfectly flat.
Even on the flat plane, curved coordinates make the connection nontrivial — Christoffel symbols can be nonzero without any real curvature.
Metric-compatible plus torsion-free pins down the connection uniquely; relax either condition and other connections appear. Do not read nonzero Christoffel symbols as proof of curvature — they can come purely from a curved coordinate choice.