Riemann curvature tensor

The tensor that measures how covariant derivatives fail to commute, encoding the intrinsic curvature of a manifold and its effects on vectors and geodesics.

Covariant derivative: A derivative that accounts for how vector and tensor components change across a curved space or varying frame. The tensor is defined by the commutator of covariant derivatives.
Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space. The tensor is defined on a manifold with a connection, commonly one induced by this metric.
Gaussian curvature: The intrinsic curvature of a two-dimensional surface at a point, equal to the product of its principal curvatures. On surfaces, the full curvature information reduces to this scalar.
Einstein field equations: The equations of general relativity relating spacetime geometry to energy and momentum. Their geometric side is built from contractions of the Riemann tensor.
Bernhard Riemann: A German mathematician whose work founded modern geometry and made curvature central to the study of manifolds. His 1854 habilitation lecture introduced the geometric framework from which the tensor developed.
Parallel transport: The rule for carrying vectors along a curve while keeping them covariantly constant. Transport around a closed loop reveals curvature through the vector's change.
Affine connection: A rule for differentiating vector fields along directions on a manifold. A connection supplies the covariant derivatives whose commutator defines curvature.
Ricci curvature: A tensor obtained by contracting the Riemann curvature tensor, measuring curvature averaged over directions. It retains part of the tensor's information while discarding directional detail.
Gravitational tidal force: The relative acceleration of nearby bodies caused by spatial variation in a gravitational field. In general relativity, the curvature tensor governs tidal acceleration.
Riemannian geometry: The study of smooth manifolds equipped with inner products on their tangent spaces. The tensor is a central invariant of this geometric theory.

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