Curvature

The curvature of a connection measures the failure of parallel transport to be path-independent — equivalently, the failure of covariant derivatives to commute. It is the geometric identity of the gauge field strength $F_{\mu \nu }$, and its integrals are the topological charges of later pages. This page builds directly on connections.md.

The curvature 2-form

The curvature of a connection with 1-form $\omega$ (or local potential $A$) is the $\mathfrak{g}$-valued 2-form $$\boxed{\Omega =d\omega +\frac{1}{2}[\omega ,\omega ],F=dA+A\wedge A}$$ (the Cartan structure equation). In components, using $A=A_{\mu }^a{T}^{a}d{x}^{\mu }$ and $[T^a,T^b]=i{f}^{abc}{T}^c$, $$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }],F_{\mu \nu }^c={\partial }_{\mu }{A}_{\nu }^c-{\partial }_{\nu }{A}_{\mu }^c-f^{abc}{A}_{\mu }^a{A}_{\nu }^b.$$ This is exactly the Yang–Mills field strength of gauge/yang-mills.md § field strength. The nonlinear $A\wedge A$ term is the source of the gluon/W self-interaction; it vanishes for abelian $U(1)$, recovering $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$, the electromagnetic field-strength tensor.

Curvature as non-commuting covariant derivatives

Equivalently, curvature is the commutator of covariant derivatives: $$[{\nabla }_{\mu },{\nabla }_{\nu }]\psi =F_{\mu \nu }\psi .$$ Covariant derivatives fail to commute by exactly the curvature — the operational meaning of "the connection is not flat". This is the same structure as the Riemann tensor in general relativity, where $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\rho }=R^{\rho }{}_{\sigma \mu \nu }{V}^{\sigma }$ (see riemannian-bridge.md).

Gauge covariance

Unlike the potential $A$, the curvature transforms homogeneously (covariantly) under a gauge change $g$: $$F\mapsto g^{-1}Fg.$$ So $F$ is a genuine section of the adjoint bundle (associated-bundles.md) — no inhomogeneous $g^{-1}dg$ term. Consequently gauge-invariant quantities are built from traces: $\mathrm{tr}(F\wedge \star F)$ is the Yang–Mills action, and $\mathrm{tr}(F\wedge F)$ is the topological density of characteristic-classes.md. For abelian $F$, which is gauge-invariant outright, $F$ itself is observable.

The Bianchi identity

Differentiating the structure equation gives the Bianchi identity $$d_{\nabla }F\equiv dF+[A,F]=0,$$ an identity (true for every connection), not a field equation. In components it is the cyclic ${\nabla }_{[\lambda }{F}_{\mu \nu ]}=0$. For electromagnetism it is the homogeneous pair of Maxwell's equations ($\nabla \cdot \vec{B}=0$, ${\partial }_t\vec{B}+\nabla \times \vec{E}=0$); for gravity it is the Bianchi identity of the Riemann tensor that underlies energy–momentum conservation. The inhomogeneous equation ($d\star F=\star J$, the Yang–Mills equation of motion) is by contrast dynamical — it comes from varying the action, not from geometry.

Flatness and integrability

A connection is flat if $F=0$. Then:

• parallel transport is locally path-independent (locally pure gauge, $A=g^{-1}dg$), by the Frobenius theorem applied to the horizontal distribution;
• but globally it can still have nontrivial holonomy around non-contractible loops — the Aharonov–Bohm effect (holonomy.md), classified by the fundamental group and $H^1$ (de-rham.md).

So $F=0$ kills the local geometry but not the global topology — the precise separation of curvature (local) from holonomy/characteristic classes (global) that organizes the rest of this folder.

References

• Kobayashi & Nomizu, Foundations of Differential Geometry, Vol. 1, Ch. 2–3.
• Nakahara, Geometry, Topology and Physics, Ch. 10.
• Baez & Muniain, Gauge Fields, Knots and Gravity, Part II.