Curvature of Connections

The curvature of a connection measures the extent to which parallel transport around infinitesimal loops fails to return to the identity. It is the fundamental local invariant of a connection.

Curvature as a 2-Form

Definition7.5Curvature of a connection

Let $\nabla$ be a connection on a vector bundle $E \to M$ . The curvature is the $\mathrm{End}(E)$ -valued 2-form $F^\nabla \in \Omega^2(M; \mathrm{End}(E))$ defined by

F^\nabla(X, Y)s = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X,Y]} s,

for vector fields $X, Y$ and section $s$ . The curvature is $C^\infty(M)$ -linear in all arguments, confirming it is a tensor.

Definition7.6Curvature in local coordinates

In a local frame $\{e_i\}$ with connection forms $\omega = (\omega^j_i)$ , the curvature form is the matrix-valued 2-form

\Omega^j_i = d\omega^j_i + \omega^j_k \wedge \omega^k_i,

or in matrix notation: $\Omega = d\omega + \omega \wedge \omega$ . This is the structure equation (Cartan's second structural equation).

The Bianchi Identity

Theorem7.2Bianchi identity

For any connection $\nabla$ with curvature $F$ :

d^\nabla F = 0,

where $d^\nabla$ is the covariant exterior derivative. In local coordinates: $d\Omega + \omega \wedge \Omega - \Omega \wedge \omega = 0$ , or $d\Omega = \Omega \wedge \omega - \omega \wedge \Omega$ .

Proof

Compute directly:

d\Omega = d(d\omega + \omega \wedge \omega) = d\omega \wedge \omega - \omega \wedge d\omega.

Also $\omega \wedge \Omega = \omega \wedge d\omega + \omega \wedge \omega \wedge \omega$ and $\Omega \wedge \omega = d\omega \wedge \omega + \omega \wedge \omega \wedge \omega$ . Thus $d\Omega + \omega \wedge \Omega - \Omega \wedge \omega = d\omega \wedge \omega - \omega \wedge d\omega + \omega \wedge d\omega + \omega^3 - d\omega \wedge \omega - \omega^3 = 0$ . $\blacksquare$

■

Flat Connections

Definition7.7Flat connection

A connection is flat if its curvature vanishes identically: $F^\nabla = 0$ . Equivalently, parallel transport depends only on the homotopy class of the path (not the path itself).

ExampleFlat connections and representations

Flat connections on a vector bundle $E \to M$ are in bijection with representations $\pi_1(M) \to \mathrm{GL}(r, \mathbb{R})$ (the monodromy representation). This is because a flat connection allows consistent parallel transport around loops, and the resulting holonomy depends only on the homotopy class of the loop.

RemarkGauge transformations

A gauge transformation is a bundle automorphism $\Phi: E \to E$ covering the identity on $M$ . Under $\Phi$ , the connection transforms as $\nabla \mapsto \Phi^{-1} \circ \nabla \circ \Phi$ , and the curvature transforms as $F \mapsto \Phi^{-1} F \Phi$ (conjugation). In local frames (physics notation): $A \mapsto g^{-1}Ag + g^{-1}dg$ and $F \mapsto g^{-1}Fg$ . The curvature is a gauge-covariant quantity.

RemarkConnections in physics

In gauge theory (Yang-Mills theory), a connection on a principal $G$ -bundle is a gauge field ( $A_\mu$ in physics notation), and the curvature is the field strength ( $F_{\mu\nu}$ ). Maxwell's electromagnetism corresponds to $G = U(1)$ , and the Standard Model uses $G = SU(3) \times SU(2) \times U(1)$ . The Yang-Mills equations $d^\nabla *F = 0$ generalize Maxwell's equations.