Curvature shape

In differential geometry, the curvature form is a generalization of the curvature tensor to a principal bundle with arbitrary connection.

Sea E → B a fiber with structure group the group of Lie G and ${displaystyle {mathfrak {g}}}}$ Lie algebra G.

We assume that ω denotes the 1-form to values in ${displaystyle {mathfrak {g}}}}$ which defines the connection in a fiberdo. Then the shape of curvature is the 2-form Ω= d ω + ω ∧ ω here. d is the external derivative and ∧ is the wedge product (it is a bit strange to apply the wedge product to the shapes with values in ${displaystyle {mathfrak {g}}}}$but works the same way.

For tangent fiber from a variety of Riemann we have O(n) as the structure group and the Ω is the 2-form with values in ${displaystyle {mathfrak {o}}}$(n) (which can be thought as antisymmetric matrices, given an orthonormal basis). In this case the Ω form is an alternative description of the curvature tensor, namely in the standard notation (using a coordinated tangent frame ${displaystyle partial _{s}}$) for the curvature tensor we have

${displaystyle R(X,Y)partial _{k}={Omega ^{s}_{k}(X,Y)partial _{s}}}}$.