Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

Let G be a Lie group with Lie algebra ${\displaystyle {\mathfrak {g}}}$, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a ${\displaystyle {\mathfrak {g}}}$-valued one-form on P).

Then the curvature form is the ${\displaystyle {\mathfrak {g}}}$-valued 2-form on P defined by

(In another convention, 1/2 does not appear.) Here ${\displaystyle d}$ stands for exterior derivative, ${\displaystyle [\cdot \wedge \cdot ]}$ is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

where X, Y are tangent vectors to P.

There is also another expression for Ω: if X, Y are horizontal vector fields on P, then

where hZ means the horizontal component of Z, on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and ${\displaystyle \sigma \in \{1,2\}}$ is the inverse of the normalization factor used by convention in the formula for the exterior derivative.

A connection is said to be flat if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology.