the Levi-Civita connection
/ LAY-vee chee-VEE-tah /
Imagine walking on a curved surface carrying an arrow and trying to keep it 'pointing the same way' as you go. On a flat plane this is obvious; on a sphere it is not, because there is no global notion of 'the same direction' far away. The Levi-Civita connection is the precise rule for transporting and differentiating vectors that the metric forces upon you — the canonical way to compare tangent vectors at nearby points so that lengths and angles are preserved and nothing is artificially twisted.
It is the unique connection nabla guaranteed by the fundamental theorem: metric-compatible (nabla g = 0) and torsion-free (nabla_X Y - nabla_Y X = [X, Y]). Concretely, nabla_X Y is a new vector field, the covariant derivative of Y in the direction X. In coordinates its components are governed by the Christoffel symbols Gamma^k_ij via nabla_(d/dx^i) (d/dx^j) = Gamma^k_ij (d/dx^k). The associated parallel transport moves a vector along a curve while keeping nabla of it zero along the curve; because nabla is metric-compatible, parallel transport is an isometry between tangent spaces, preserving lengths and angles.
Where it shows up: geodesics are curves whose velocity is parallel to itself (nabla_(gamma') gamma' = 0); curvature is the failure of second covariant derivatives to commute; and all of intrinsic Riemannian geometry is built on nabla. A subtle point worth keeping straight: the Levi-Civita / Riemannian exponential map exp_p (built from geodesics of nabla) is a DIFFERENT construction from the Lie-group exponential exp: g -> G, and they coincide only for bi-invariant metrics on a Lie group. Do not conflate the two.
Parallel-transport a tangent vector around a spherical triangle (e.g. start at the north pole pointing toward a meridian, slide to the equator, along it, and back up). When you return, the arrow has rotated by an angle equal to the triangle's enclosed solid angle — a visible signature of the sphere's curvature, encoded entirely in the Levi-Civita connection.
Holonomy around a loop: parallel transport with the Levi-Civita connection rotates a vector by an amount measuring enclosed curvature.
The Riemannian exponential map exp_p and the Lie-group exponential exp: g -> G are different constructions; they agree only for bi-invariant metrics. Never assume a 'connection' on a manifold is Levi-Civita unless metric-compatibility and torsion-freeness are stated.