the Christoffel symbols
/ KRIS-tof-el /
When you write the rule for differentiating vectors (the Levi-Civita connection) in actual coordinates, it spits out a set of numbers at every point that say how the coordinate directions twist as you move. Those numbers are the Christoffel symbols. They are the bookkeeping that corrects the naive derivative: if your coordinate grid bends or stretches, the symbols quietly add the right adjustment so the derivative tracks the true geometric change rather than an artifact of the grid.
Precisely, in coordinates x^1, ..., x^n the Christoffel symbols Gamma^k_ij record how the basis vector d/dx^i changes in the direction d/dx^j: nabla_(d/dx^i) (d/dx^j) = sum_k Gamma^k_ij d/dx^k. For the Levi-Civita connection they are computed entirely from the metric by an explicit formula: Gamma^k_ij = (1/2) sum_l g^kl (d_i g_jl + d_j g_il - d_l g_ij), where g^kl is the inverse metric matrix and d_i means the partial derivative with respect to x^i. To use them, you plug them into the geodesic equation or the covariant derivative; they are the coordinates' contribution to every curved-space computation.
Two honest cautions keep students out of trouble. First, the Christoffel symbols are not the components of a tensor: they transform with an extra inhomogeneous term under a change of coordinates, which is exactly why you can make them vanish at any single point by choosing good coordinates, yet you cannot make them vanish everywhere unless the space is genuinely flat. Second, they depend on the coordinate system, so nonzero Christoffel symbols do not by themselves signal curvature — polar coordinates on the flat plane already have nonzero ones. True curvature lives in a particular combination of the symbols and their derivatives, namely the Riemann tensor.
On the flat plane in polar coordinates (r, theta), the metric is ds^2 = dr^2 + r^2 dtheta^2. Working through the formula gives nonzero symbols Gamma^r_(theta theta) = -r and Gamma^theta_(r theta) = Gamma^theta_(theta r) = 1/r. These are exactly the corrections that keep a straight line straight when described in polar coordinates — pure coordinate effects, since the plane is flat.
Polar coordinates on the flat plane already produce nonzero Christoffel symbols; they record coordinate twisting, not curvature.
Christoffel symbols are not tensor components and can always be zeroed at a single chosen point (normal coordinates). So they are never a coordinate-independent measure of curvature — that role belongs only to the Riemann tensor built from them.