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