# Connection form

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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 ; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev . 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 .

How to Cite This Entry:
Connection form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_form&oldid=13808
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article