Exterior Covariant Derivative

Let ''P'' → ''M'' be a Principal ''G''-bundle on a Smooth Manifold ''M''. If

\phi

is a Tensorial ''k''-form on ''P'', then its exterior covariant derivative is defined by
:

D\phi(X_0,X_1,\dots,X_k)=\mathrm{d}\phi(h(X_0),h(X_1),\dots,h(X_k))

where ''h'' denotes the projection to the horizontal subspace,

H_x

defined by the connection, with kernel

V_x

(the vertical subspace) of the tangent bundle of the Total Space of the Fiber Bundle . Here

X_i

are any vector fields on ''P''. ''D''φ is a tensorial ''k''+1 form on ''P''.

Unlike the usual Exterior Derivative , which squares to 0, we have
:

D^2\phi=\Omega\wedge\phi

where

\Omega

denotes the Curvature Form . In particular

D^2

vanishes for a Flat Connection .

REFERENCES

	CATEGORIES ABOUT EXTERIOR COVARIANT DERIVATIVE

	connection mathematics

	differential geometry

	fiber bundles

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