Exterior covariant derivative
Encyclopedia
In mathematics
, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds
, which makes it possible to simplify formulas which use a principal connection
.
Let P → M be a principal G-bundle
on a smooth manifold M. If
is a tensorial k-form on P, then its exterior covariant derivative is defined by
where h denotes the projection to the horizontal subspace,
defined by the connection, with kernel 
(the vertical subspace) of the tangent bundle of the total space of the fiber bundle
. Here
are any vector fields on P. Dφ is a tensorial k+1 form on P.
Unlike the usual exterior derivative
, which squares to 0, we have
where
denotes the curvature form
. In particular
vanishes for a flat connection.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds
Calculus on Manifolds
Michael Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus is a short text treating analysis in several variables in Euclidean spaces and on differentiable manifolds...
, which makes it possible to simplify formulas which use a principal connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
.
Let P → M be a principal G-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
on a smooth manifold M. If
is a tensorial k-form on P, then its exterior covariant derivative is defined bywhere h denotes the projection to the horizontal subspace,
defined by the connection, with kernel (the vertical subspace) of the tangent bundle of the total space of the fiber bundleFiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
. Here
are any vector fields on P. Dφ is a tensorial k+1 form on P.Unlike the usual exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
, which squares to 0, we have
where
denotes the curvature formCurvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
. In particular
vanishes for a flat connection.