Christoffel symbols

Christoffel symbols are coordinate-dependent coefficients describing how coordinate basis vectors change across a manifold. They encode a connection in a chosen coordinate system, rather than defining a tensor themselves.

Levi-Civita connection: The unique torsion-free connection compatible with a given metric. Its coefficients are the Christoffel symbols most often computed in Riemannian geometry.
Manifold: A space that locally resembles Euclidean space and supports smooth coordinate descriptions. Christoffel symbols describe connections on manifolds in local coordinates.
General relativity: A theory of gravitation in which spacetime geometry is dynamical and matter shapes curvature. They enter coordinate equations for freely falling bodies and spacetime curvature.
Tensor: A multilinear geometric object whose components transform tensorially between coordinate systems. Christoffel symbols fail the tensor transformation law despite carrying multiple indices.
Covariant derivative: A derivative that accounts for changes in geometric structure across a manifold. Christoffel symbols supply the correction terms that make coordinate derivatives geometrically meaningful.
Tangent space: The vector space of tangent directions at a point on a smooth manifold. Each coordinate chart supplies a basis in each tangent space whose variation the symbols encode.
Riemann curvature tensor: A tensor measuring the curvature of a connection through the failure of covariant derivatives to commute. Its coordinate formula combines derivatives and products of Christoffel symbols.
Spin connection: A connection acting on spinor fields, commonly expressed in an orthonormal frame. It plays a connection role like Christoffel symbols but acts on frame-based spinors.
Metric tensor: A tensor that defines inner products on tangent spaces and determines lengths, angles, and distances. For the Levi-Civita connection, derivatives of the metric determine the coefficients.
Coordinate chart: A map assigning local coordinates to points of a manifold. The symbols depend on the chart used to express the connection.

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