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 \({v}\))
- \({\parallel^{\lambda}{}_{\mu}\left(C\right)w^{\mu}=P\textrm{exp}\left(-\int_{C}\Gamma^{\lambda}{}_{\mu}\right)w^{\mu}}\)