Bianchi identities

Differential identities obeyed by curvature tensors. In general relativity, their contracted form implies that the Einstein tensor has zero covariant divergence.

Riemann curvature tensor: A tensor measuring the failure of covariant derivatives to commute and encoding the curvature of a space. The identities express differential constraints on this curvature tensor.
Affine connection: A structure that defines parallel transport and covariant differentiation on a manifold. The identities apply to curvature defined by a connection, not only to a particular coordinate formula.
Einstein field equations: Equations relating spacetime curvature to the energy and momentum of matter and radiation. Their geometric side has zero divergence, requiring compatible conservation of the matter side.
Luigi Bianchi: An Italian mathematician whose work on differential geometry gave the Bianchi identities their name. His studies of geometric curvature established the identities’ historical association with his name.
Covariant derivative: A derivative that accounts for changes in geometric or tensorial structure across a curved space. Its noncommuting action underlies the curvature relation in the differential identity.
Ricci tensor: A contraction of the Riemann curvature tensor that summarizes curvature in each direction. Its divergence is related to the gradient of scalar curvature by the contracted identity.
Stress-energy tensor: A tensor describing the density and flux of energy and momentum in physical systems. The field equations and Bianchi identity imply its covariant conservation when the cosmological constant is fixed.
Elwin Bruno Christoffel: A German mathematician who introduced connection coefficients now called Christoffel symbols. His coordinate framework helped make curvature relations expressible in differential geometry.
Christoffel symbols: Connection coefficients that describe covariant differentiation in a chosen coordinate system. Their derivative and quadratic terms yield the coordinate form of the identities.
Scalar curvature: A scalar obtained by contracting the Ricci tensor, representing a pointwise summary of curvature. Its gradient appears alongside the Ricci tensor’s divergence in the contracted identity.

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