Affine connection

An affine connection is a rule for differentiating vector fields along vector fields on a manifold. It also defines parallel transport and, through its failure to commute, curvature.

Covariant derivative: A derivative that accounts for how geometric or algebraic structure varies across a space. An affine connection is the rule that supplies covariant derivatives of vector fields.
Smooth manifold: A space locally modeled on Euclidean space, with smoothly compatible coordinate charts. Affine connections differentiate fields on this underlying smooth space.
Directional derivative: The rate of change of a function or vector-valued quantity in a specified direction in a fixed setting. Unlike ordinary directional differentiation, a connection compares tangent vectors at different manifold points.
Riemannian geometry: The study of smooth manifolds equipped with smoothly varying inner products on tangent spaces. Its metric determines a canonical torsion-free affine connection.
Parallel transport: A method for moving vectors along curves while preserving a chosen notion of parallelism. The connection specifies how vectors change when transported along a curve.
Tangent bundle: The bundle whose fiber at each point of a manifold is its tangent vector space. The connection acts on sections of the tangent bundle.
Levi-Civita connection: The unique torsion-free connection compatible with a given Riemannian metric. It is the distinguished affine connection selected when a metric is supplied.
General relativity: A theory of gravitation in which spacetime geometry is dynamical and shaped by matter and energy. Its Levi-Civita connection describes free fall and enters the field equations.
Curvature tensor: A tensor measuring the failure of covariant derivatives to commute. Commutators of connection derivatives define the connection's curvature.
Vector field: A smooth assignment of a tangent vector to each point of a manifold. Vector fields are both the inputs to and outputs of the connection's derivative.

Linked from 12 pages