The connection

If we view \({\nabla}\) as a map from two vector fields \({v}\) and \({w}\) to a third vector field \({\nabla_{v}w}\), it is called an affine connection. Note that since no use has been made of coordinates or frames in the definition of \({\nabla}\), it is a frame-independent quantity.

Since \({\nabla_{v}}\) is linear in \({v}\), and depends only on its local value, we can regard \({\nabla}\) as a 1-form on \({M}\). If we choose a frame \({e_{\mu}}\) on \({M}\) with corresponding dual frame \({\beta^{\mu}}\), we can define the connection 1-form

\(\displaystyle \Gamma^{\lambda}{}_{\mu}\left(v\right)\equiv\beta^{\lambda}\left(\nabla_{v}e_{\mu}\right). \)

\({\Gamma^{\lambda}{}_{\mu}\left(v\right)}\) is the \({\lambda^{\textrm{th}}}\) component of the difference between the frame \({e_{\mu}}\) and its parallel transport in the direction \({v}\).

From its definition, it is clear that \({\Gamma^{\lambda}{}_{\mu}}\) is a frame-dependent object. Under a change of frame \({(\gamma^{-1})^{\nu}{}_{\mu}}\), it is not hard to see that the connection 1-form transforms as

\(\displaystyle \Gamma^{\prime\sigma}{}_{\tau}=\gamma^{\sigma}{}_{\lambda}\Gamma^{\lambda}{}_{\mu}(\gamma^{-1})^{\mu}{}_{\tau}+\gamma^{\sigma}{}_{\lambda}\mathrm{d}(\gamma^{-1})^{\lambda}{}_{\tau}, \)

where the exterior derivative \({\mathrm{d}}\) operates on each of the matrix components \({(\gamma^{-1})^{\lambda}{}_{\tau}}\) as a 0-form. This affine mapping (inhomogeneous transformation) under a change of frame, along with the way that the connection allows us to view an infinitesimal area as an affine space, explains the name “affine connection.” It also demonstrates that \({\Gamma^{\lambda}{}_{\mu}}\) cannot be viewed as the components of a tensor, as expected since it is formed from the derivative of the frame.

At a point \({p}\), the value of \({\Gamma^{\lambda}{}_{\mu}\left(v\right)}\) is an infinitesimal linear transformation on \({T_{p}M}\), so that \({\Gamma^{\lambda}{}_{\mu}}\) is a frame-dependent \({gl\left(n,\mathbb{R}\right)}\)-valued 1-form. Recalling our notation for algebra-valued forms, we can then write

\(\displaystyle \check{\Gamma}\left(v\right)\vec{w}\equiv\Gamma^{\lambda}{}_{\mu}\left(v\right)w^{\mu}e_{\lambda}=\left(\nabla_{v}e_{\mu}\right)w^{\mu}, \)

where we view \({\vec{w}}\) as a \({\mathbb{R}^{n}}\)-valued 0-form. The vector \({\check{\Gamma}\left(v\right)\vec{w}}\) measures the difference between the frame and its parallel transport in the direction \({v}\), weighted by the components of \({w}\).

Δ It is important to remember that \({\check{\Gamma}\left(v\right)\vec{w}}\) is related to the difference between the frame and its parallel transport, while \({\nabla_{v}w}\) measures the difference between \({w}\) and its parallel transport; thus unlike \({\nabla_{v}w}\), \({\check{\Gamma}\left(v\right)\vec{w}}\) depends only upon the local value of \({w}\), but takes values that are frame-dependent.
Δ Since we have used the frame to view \({\check{\Gamma}}\) as a \({gl\left(n,\mathbb{R}\right)}\)-valued 1-form, i.e. a matrix-valued 1-form, \({\vec{w}}\) must be viewed as a frame-dependent column vector of components. We could instead view \({\check{\Gamma}}\) as a \({gl\left(\mathbb{R}^{n}\right)}\)-valued 1-form and \({\vec{w}}\) as a frame-independent intrinsic vector. In this case the action of \({\check{\Gamma}}\) on \({\vec{w}}\) would be frame-independent, but the value of \({\check{\Gamma}}\) itself would remain frame-dependent. We choose to use matrix-valued forms due to the need in the next section and beyond to take the exterior derivative of component functions, but the abstract viewpoint is important to keep in mind when generalizing to fiber bundles.

Using this notation, we can view a change of frame as a (frame-dependent) \({GL\left(n,\mathbb{R}\right)}\)-valued 0-form and write the transformation of the connection 1-form under a change of frame as

\(\displaystyle \check{\Gamma}^{\prime}=\check{\gamma}\check{\Gamma}\check{\gamma}^{-1}+\check{\gamma}\mathrm{d}\check{\gamma}^{-1}. \)

An Illustrated Handbook