Summary

In general, a “manifold with connection” is one with an additional structure that “connects” the different tangent spaces of the manifold to one another in a linear fashion. Specifying any one of the above connection quantities, the covariant derivative, or the parallel transporter equivalently determines this structure. The following tables summarize the situation.

Construct Argument(s) Value Dependencies
\({\parallel_{C}}\) \({v\in T_{p}M}\) \({\parallel_{C}\left(v\right)\in T_{q}M}\) Path \({C}\) from \({p}\) to \({q}\)
\({\parallel^{\lambda}{}_{\mu}}\) Path \({C}\) \({\parallel^{\lambda}{}_{\mu}\left(C\right)\in GL}\) Frame on \({M}\)
\({\nabla_{v}}\) \({w\in TM}\) \({\nabla_{v}w\in T_{p}M}\) \({v\in T_{p}M}\)
\({\nabla}\) \({v\in T_{p}M}\), \({w\in TM}\) \({\nabla_{v}w\in T_{p}M}\) None
\({\Gamma^{\lambda}{}_{\mu}}\) \({v\in T_{p}M}\) \({\Gamma^{\lambda}{}_{\mu}\left(v\right)\in gl}\) Frame on \({M}\)
\({\check{\Gamma}\left(v\right)}\) \({\vec{w}\in T_{p}M}\) \({\check{\Gamma}\left(v\right)\vec{w}\in T_{p}M}\) Frame on \({M}\), \({v\in T_{p}M}\)
\({\Gamma^{\lambda}{}_{\mu\sigma}}\) None Connection coefficient Frame on \({M}\)

Note: Each construct above is considered at a point \({p}\); to determine a manifold with connection it must be defined for every point in \({M}.\)

Below we review the intuitive meanings of the various vector derivatives.

Vector derivative Meaning
\({L_{v}w\equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\left(w\left|_{p+\varepsilon v}\right.-\mathrm{d}\Phi_{\varepsilon}\left(w\left|_{p}\right.\right)\right)/\varepsilon}\) The difference between \({w}\) and its transport by the local flow of \({v}\).
\({\nabla_{v}w\equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\left(w\left|_{p+\varepsilon v}\right.-\parallel_{C}\left(w\left|_{p}\right.\right)\right)/\varepsilon}\) The difference between \({w}\) and its parallel transport in the direction \({v}\).
\({\frac{\mathrm{D}}{\mathrm{d}t}w\equiv\mathrm{D}_{t}w\equiv\nabla_{C^{\prime}(t)}w}\) The difference between \({w}\) and its parallel transport in the direction tangent to \({C(t)}\).
\({\Gamma^{\lambda}{}_{\mu}\left(v\right)\equiv\beta^{\lambda}\left(\nabla_{v}e_{\mu}\right)}\) The \({\lambda^{\textrm{th}}}\) component of the difference between \({e_{\mu}}\) and its parallel transport in the direction \({v}\).
\({\check{\Gamma}\left(v\right)\equiv\nabla_{v}\left(T_{p}M\right)}\) The infinitesimal linear transformation on the tangent space that takes the parallel transported frame to the frame in the direction \({v}\).
\({\check{\Gamma}\left(v\right)\vec{w}\equiv\Gamma^{\lambda}{}_{\mu}\left(v\right)w^{\mu}e_{\lambda}=\left(\nabla_{v}e_{\mu}\right)w^{\mu}}\) The difference between the frame and its parallel transport in the direction \({v}\), weighted by the components of \({w}\).
\({\Gamma^{\lambda}{}_{\mu\sigma}\equiv\Gamma^{\lambda}{}_{\mu}\left(e_{\sigma}\right)=\beta^{\lambda}\left(\nabla_{\sigma}e_{\mu}\right)}\) The \({\lambda^{\textrm{th}}}\) component of the difference between \({e_{\mu}}\) and its parallel transport in the direction \({e_{\sigma}}\).
\({\mathrm{d}\vec{w}\left(v\right)\equiv\mathrm{d}w^{\mu}\left(v\right)e_{\mu}}\) The change in the frame-dependent components of \({w}\) in the direction \({v}\).
\({\partial_{a}w^{b}\equiv\mathrm{d}w^{b}(e_{a})}\) The change in the \({b^{\mathrm{th}}}\) frame-dependent component of \({w}\) in the direction \({e_{a}}\).
\({\nabla_{a}w^{b}\equiv(\nabla_{e_{a}}w)^{b}}\) The \({b^{\mathrm{th}}}\) component of the difference between \({w}\) and its parallel transport in the direction \({e_{a}}\).

Other quantities in terms of the connection:

  • \({\nabla_{v}w=\mathrm{d}\vec{w}\left(v\right)+\check{\Gamma}\left(v\right)\vec{w}}\)
  • \({\nabla_{a}w^{b}=\partial_{a}w^{b}+\Gamma^{b}{}_{ca}w^{c}}\)
  • \({\parallel^{\lambda}{}_{\mu}\left(C\right)w^{\mu}=w^{\lambda}-\varepsilon\Gamma^{\lambda}{}_{\mu}\left(v\right)w^{\mu}}\)     (for infinitesimal \({C}\) with tangent \({v}\))
  • \({\parallel^{\lambda}{}_{\mu}\left(C\right)w^{\mu}=P\textrm{exp}\left(-\int_{C}\Gamma^{\lambda}{}_{\mu}\right)w^{\mu}}\)

An Illustrated Handbook