the Christoffel symbols
/ KRIS-toff-el /
When you write the Levi-Civita connection in a coordinate chart, you need numbers that tell you how the coordinate basis vectors twist as you move. Those numbers are the Christoffel symbols: the explicit coordinate components of the covariant derivative. They are the bookkeeping that turns the abstract connection into formulas you can actually compute geodesics and curvature with.
In coordinates x^1, ..., x^n the Christoffel symbols Gamma^k_ij are defined by nabla_(d/dx^i)(d/dx^j) = Gamma^k_ij (d/dx^k). For the Levi-Civita connection they are computed directly from the metric by Gamma^k_ij = (1/2) g^kl ( d_i g_jl + d_j g_il - d_l g_ij ), where g^kl is the inverse metric matrix, d_i means partial derivative with respect to x^i, and repeated indices are summed. They are symmetric in the lower pair, Gamma^k_ij = Gamma^k_ji, precisely because the connection is torsion-free. To use them: feed the metric coefficients in, differentiate, contract with the inverse metric, and out come the Gamma's.
Crucial honesty: Christoffel symbols are NOT the components of a tensor. Under a change of coordinates they pick up an inhomogeneous (second-derivative) term, so they can be made to vanish at any single point (normal coordinates) yet be nonzero in another chart. This is why curvature, which IS a tensor, is built from derivatives and products of the Gamma's arranged so the non-tensorial parts cancel. Two textbooks may also order or sign the indices differently, so always check a source's convention before trusting a Christoffel formula.
For polar coordinates on the flat plane, ds^2 = dr^2 + r^2 dtheta^2, the nonzero Christoffel symbols are Gamma^r_(theta theta) = -r and Gamma^theta_(r theta) = Gamma^theta_(theta r) = 1/r. Plugging these into the geodesic equation recovers that straight lines, written in polar form, are the geodesics — even though the plane is flat and a Cartesian chart would have all Gamma's zero.
Nonzero Christoffel symbols on flat space are an artifact of curved coordinates, not of curvature itself.
Christoffel symbols are not tensor components: they can be nonzero on flat space (curved coordinates) and zeroed at any point (normal coordinates). Curvature, built from them so non-tensorial terms cancel, is the genuine coordinate-independent quantity.