Covariant derivative

A derivative of tensor fields that accounts for changes in geometric bases across a manifold. It extends ordinary differentiation while preserving tensorial meaning across curved spaces.

Connection (differential geometry): A structure that defines how to compare vectors in neighboring tangent spaces and differentiate fields on a manifold. A connection supplies the general rule from which covariant differentiation is constructed.
Tangent space: A vector space of tangent directions at a point of a smooth manifold. Covariant differentiation compares tensor data attached to these pointwise vector spaces.
Riemann curvature tensor: A tensor measuring the failure of covariant derivatives to commute on a manifold. Commuting this derivative in different directions produces the curvature tensor.
Ordinary derivative: The rate of change of a function or coordinate expression with respect to a variable. It suffices for scalar functions but does not by itself differentiate geometric fields invariantly.
Christoffel symbols: Coordinate-dependent coefficients that describe a connection and appear in coordinate formulas for covariant differentiation. They correct ordinary component derivatives for the changing coordinate basis.
Tensor field: A smoothly varying assignment of a tensor to each point of a manifold. These are the fields whose derivatives the operation defines.
Geodesic: A curve whose tangent vector is parallel transported along itself under a connection. Its defining equation sets the covariant derivative of the tangent along the curve to zero.
Lie derivative: A derivative measuring how a tensor field changes along the flow of a vector field. It compares fields by flow rather than by a connection's parallel transport.
Parallel transport: A rule for carrying vectors along curves while keeping them parallel according to a connection. The connection defining this derivative also determines which vectors count as parallel along a path.
Directional derivative: The rate of change of a function or field along a specified direction. Covariant differentiation generalizes directional differentiation from scalar functions to geometric fields.

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