the Christoffel symbols
/ kris-TOF-el /
When you carry a compass across a curved surface, or simply switch to curved coordinates, the very grid lines you measure against twist and bend from point to point. Differentiating a vector then means comparing it against a moving frame, and you need bookkeeping to keep track of how the frame itself changes. The Christoffel symbols are exactly that bookkeeping: at each point they record how the coordinate basis vectors rotate and stretch as you step to a neighbouring point.
They are computed entirely from the metric and its first derivatives: Gamma^a_bc = (1/2) g^ad (partial_b g_dc + partial_c g_db - partial_d g_bc). They are symmetric in the two lower indices, b and c, which is what it means for the connection to be torsion-free (the standard Levi-Civita connection of general relativity). Despite carrying indices, they are NOT the components of a tensor: under a change of coordinates they pick up an extra inhomogeneous term, which is precisely why they can encode the fictitious, coordinate-dependent part of a gravitational field.
You meet them everywhere in the machinery of curved space: inside the geodesic equation, inside the covariant derivative, and as the raw material from which the Riemann curvature tensor is assembled. The single most important caveat follows from their not being a tensor: Christoffel symbols can be nonzero in one coordinate system and made to vanish at a point by switching to a locally inertial frame. Nonzero Gamma therefore does not signify curvature; genuine, frame-independent curvature only shows up in the Riemann tensor.
Even in flat two-dimensional space, using polar coordinates gives nonzero Christoffel symbols such as Gamma^r_(theta theta) = -r and Gamma^theta_(r theta) = 1/r. They encode nothing but the curving of the coordinate grid, since the plane itself is perfectly flat.
Nonzero Christoffel symbols in flat space: coordinates, not curvature.
Because they are not a tensor, you cannot ask whether Christoffel symbols 'really' vanish; the answer depends on the coordinates. Only tensorial statements, like whether the Riemann tensor is zero, are frame-independent facts about the geometry.