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.

ConstructArgument(s)ValueDependencies
\({\parallel_{C}}\)\({v\in T_{p}M}\)\({\parallel_{C}\left(v\right)\in T_{q}M}\)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 \({M}\), \({v\in T_{p}M}\)
\({\Gamma^{\lambda}{}_{\mu\sigma}}\)NoneConnection coefficientFrame 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 derivativeMeaning
\({L_{v}w\equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\left(w\left|_{p+\varepsilon v}\right.-\mathrm{d}\Phi_{\varepsilon}\left(w\left|_{p}\right.\right)\right)/\varepsilon}\)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