The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative.
In other terms,[1] 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[2] 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
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.
If E → B is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan: where
is a usual 2-form) then For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices.
In this case the form Ω is an alternative description of the curvature tensor, i.e. using the standard notation for the Riemannian curvature tensor.
is the canonical vector-valued 1-form on the frame bundle, the torsion
is the vector-valued 2-form defined by the structure equation where as above D denotes the exterior covariant derivative.
The first Bianchi identity takes the form The second Bianchi identity takes the form and is valid more generally for any connection in a principal bundle.
The Bianchi identities can be written in tensor notation as:
The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of general theory of relativity.