Connection form

A linear differential form on a principal fibre bundle that takes values in the Lie algebra of the structure group of . It is defined by a certain linear connection on , and it determines this connection uniquely. The values of the connection form in terms of , where and , are defined as the elements of which, under the action of on , generate the second component of relative to the direct sum . Here is the fibre of that contains and is the horizontal distribution of . The horizontal distribution , and so the connection , can be recovered from the connection form in the following way.

The Cartan–Laptev theorem. For a form on with values in to be a connection form it is necessary and sufficient that: 1) , for , is the element of that generates under the action of on ; and 2) the -valued -form

formed from , is semi-basic, or horizontal, that is, if at least one of the vectors belongs to . The -form is called the curvature form of the connection. If a basis is defined in , then condition 2) can locally be expressed by the equalities:

where are certain linearly independent semi-basic -forms. The necessity of condition 2) was established in this form by E. Cartan [1]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [2]. The equations

for the components of the connection form are called the structure equations for the connection in , the define the curvature object.

As an example, let be the space of affine frames in the tangent bundle of an -dimensional smooth manifold . Then and are, respectively, the group and the Lie algebra of matrices of the form

and

By the Cartan–Laptev theorem, the -valued -form

on is the connection form of a certain affine connection on if and only if

Here and form, respectively, the torsion and curvature tensors of the affine connection on . The last two equations for the components of the connection form are called the structure equations for the affine connection on .

References

[1]	E. Cartan, "Espaces à connexion affine, projective et conforme" Acta Math. , 48 (1926) pp. 1–42
[2]	G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)