Connection form

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.

Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.

A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.

Let ${\displaystyle E}$ be a vector bundle of fibre dimension ${\displaystyle k}$ over a differentiable manifold ${\displaystyle M}$. A local frame for ${\displaystyle E}$ is an ordered basis of local sections of ${\displaystyle E}$. It is always possible to construct a local frame, as vector bundles are always defined in terms of local trivializations, in analogy to the atlas of a manifold. That is, given any point ${\displaystyle x}$ on the base manifold ${\displaystyle M}$, there exists an open neighborhood ${\displaystyle U\subseteq M}$ of ${\displaystyle x}$ for which the vector bundle over ${\displaystyle U}$ is locally trivial, that is isomorphic to ${\displaystyle U\times \mathbb {R} ^{k}}$ projecting to ${\displaystyle U}$. The vector space structure on ${\displaystyle \mathbb {R} ^{k}}$ can thereby be extended to the entire local trivialization, and a basis on ${\displaystyle \mathbb {R} ^{k}}$ can be extended as well; this defines the local frame. (Here the real numbers are used, although much of the development can be extended to modules over rings in general, and to vector spaces over complex numbers ${\displaystyle \mathbb {C} }$ in particular.)

Let ${\displaystyle \mathbf {e} =(e_{\alpha })_{\alpha =1,2,\dots ,k}}$ be a local frame on ${\displaystyle E}$. This frame can be used to express locally any section of ${\displaystyle E}$. For example, suppose that ${\displaystyle \xi }$ is a local section, defined over the same open set as the frame ${\displaystyle \mathbb {e} }$. Then

where ${\displaystyle \xi ^{\alpha }(\mathbf {e} )}$ denotes the components of ${\displaystyle \xi }$ in the frame ${\displaystyle \mathbf {e} }$. As a matrix equation, this reads

In general relativity, such frame fields are referred to as tetrads. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold ${\displaystyle M}$ (the coordinate system on ${\displaystyle M}$ being established by the atlas).

A connection in E is a type of differential operator