the Bianchi identity
/ bee-AHN-kee /
The Bianchi identity is the automatic constraint that curvature must satisfy simply because it came from a connection. You do not impose it; it falls out the moment you differentiate the structure equation. Intuitively it says that curvature, viewed as a flux, is 'conserved' — it has no sources of its own — which is why in physics it becomes half of Maxwell's equations for free, with no field equation needed.
Precisely, for a connection omega with curvature Omega = d omega + omega ^ omega on a principal bundle, the (differential, or second) Bianchi identity states that the covariant exterior derivative of the curvature vanishes: D Omega = d Omega + [omega, Omega] = 0. The clean way to see it: apply d to the structure equation Omega = d omega + omega ^ omega and use d^2 = 0; the terms reorganize into exactly D Omega = 0. In a gauge with potential A and field strength F it reads dF + [A, F] = 0, written compactly D_A F = 0. For an abelian connection (F = dA) this collapses to dF = 0, automatic from d^2 = 0. Separately, on the tangent bundle there is also a first (algebraic) Bianchi identity, the symmetry R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0, which holds for torsion-free connections — a different statement from the differential one.
Why it matters: the Bianchi identity is what makes Chern-Weil theory work. Because the curvature is covariantly closed, the invariant polynomials of it produce honestly closed differential forms, whose de Rham classes are the characteristic classes; without D Omega = 0 those forms would not be closed and the construction would collapse. In gauge theory dF + [A,F] = 0 is the source-free half of the field equations (the Yang-Mills equation D * F = J is the dynamical other half). One caveat: keep the two Bianchi identities distinct — the algebraic first identity is a pointwise symmetry tied to vanishing torsion, while the differential second identity is the conservation law that holds for any connection's curvature.
In electromagnetism F = dA is a 2-form, and the Bianchi identity is simply dF = 0. Writing it out in components gives partial_[mu F_{nu rho]} = 0, which is precisely the homogeneous Maxwell equations: Gauss's law for magnetism (no magnetic monopoles) and Faraday's law of induction. These hold identically, with no charges or currents involved, because they are a geometric identity, not a dynamical law.
Half of Maxwell's equations is the Bianchi identity dF = 0 — geometry, not physics input.
There are two Bianchi identities and they are easy to confuse: the first (algebraic) identity R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0 is a pointwise symmetry that requires zero torsion, while the second (differential) identity D Omega = 0 is the covariant-closedness that holds for every connection. Chern-Weil relies on the second; the curvature-tensor symmetries use the first.