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A connection on a vector bundle pulls back along a smooth curve $\gamma$ to differentiate its time-dependent sections. For the tangent bundle, a section $V(t)$ of the pullback tangent bundle hasThe Leibniz rule under a change of frame proves independence of the coordinates. The derivative includes the time dependence of the coefficients, so it is meaningful even at a zero tangent velocity. A vector field along a map need not extend to one ambient field if the curve self-intersects. Parallel fields satisfy $D_{t}V=0$, a linear ordinary differential equation whose unique solutions define parallel transport.

A smooth vector field along a map $\gamma :[a,b]\to M$ is a smooth section of the pullback tangent bundle ${\gamma }^{\ast }TM$: it assigns $V(t)\in T_{\gamma (t)}M$, with smooth coefficients in each local frame. It need not be the restriction of one ambient vector field, since a curve may return to the same point with different values of $V$.

An affine connection, also called a Koszul connection, induces a pullback connection on this bundle. In coordinates, write $V=V^i(t){\partial }_i$ and ${\nabla }_{{\partial }_j}{\partial }_k={\Gamma }^i{}_{jk}{\partial }_i$. Its covariant derivative along a curve isThe connection transformation law, or the Leibniz rule applied to a change of frame, makes this expression independent of the frame. In particular it is defined even when $\dot{\gamma }=0$; it differentiates the section's coefficients as well as the frame.

The field is parallel precisely when $D_{t}V=0$. In a frame along a coordinate segment this is the linear ordinary differential equationThe Picard-Lindelof theorem gives a unique solution with prescribed initial value. Smooth coefficients on compact subintervals are bounded, and the usual norm estimate for a linear ordinary differential equation prevents finite-time blowup there. A finite sequence of coordinate segments covers the image of the compact interval $[a,b]$; solve successively and use uniqueness on overlaps. This gives a unique parallel field on the whole interval for every $V_a\in T_{\gamma (a)}M$.

Define parallel transport by ${\tau }_t(V_a)=V(t)$. Linearity of the equation and uniqueness give ${\tau }_t(c{V}_a+d{W}_a)=c{\tau }_t(V_a)+d{\tau }_t(W_a)$. Solving along the reversed curve supplies its inverse. Therefore each ${\tau }_t:T_{\gamma (a)}M\to T_{\gamma (t)}M$ is a linear isomorphism.

For two affine connections, put $\Delta (X,Y)={\nabla }_{X}Y-{\tilde{\nabla }}_{X}Y$. Their rules givesince the two $X(f)Y$ terms cancel. The tensoriality of this operation proves that the difference of affine connections is a tensor, of type $(1,2)$, with smooth coordinate coefficients ${\Gamma }^i{}_{jk}-{\tilde{\Gamma }}^i{}_{jk}$.

A parametrized geodesic satisfies $D_t\dot{\gamma }=0$, or the geodesic equationWith $v^i={\dot{\gamma }}^i$, this becomes the first-order system ${\dot{\gamma }}^i=v^i$, ${\dot{v}}^i=-{\Gamma }^i{}_{jk}(\gamma )v^j{v}^k$. Its right side is smooth, so the Picard-Lindelof theorem gives a unique local solution for each initial point and tangent vector. This establishes uniqueness with the parametrization fixed.

The two accelerations differ byIf $\Delta (v,v)=0$ for every tangent vector, either acceleration vanishes exactly when the other does. Conversely, start the $\nabla$-geodesic with arbitrary initial vector $v$ at an arbitrary point. If it is also a $\tilde{\nabla }$-geodesic with the same parameter, evaluating this identity initially gives $\Delta (v,v)=0$. HencePolarization makes the latter condition equivalent to $\Delta (u,v)+\Delta (v,u)=0$: parametrized geodesics determine the symmetric part of an affine connection. In particular torsion-free connections are determined by their parametrized geodesics, since their difference is also symmetric and must then vanish.

At $p\in \mathcal{Q}$, the differential of a smooth map defines the pushforward of a vector field byThis is a tangent vector at $\psi (p)$, and varies smoothly as a section of the pullback tangent bundle. In local coordinates its components are $X^i(p){\partial }_i{\psi }^a(p)$.

For a general smooth map between manifolds, this is a vector field along a map, rather than an intrinsic vector field on all of $N$. A vector field $Y$ on the image can be defined only if these vectors agree whenever two points have the same image, and the resulting field must also be smooth; a projectable vector field must meet that requirement. For a diffeomorphism, there is no ambiguity and $Y_q=d{\psi }_{{\psi }^{-1}(q)}{X}_{{\psi }^{-1}(q)}$. For example, $\psi (x)=x^2$ on $\mathbb{R}$ and $X={\partial }_x$ give $2x{\partial }_y$, with opposite nonzero values at the two preimages of $y>0$. Thus the source's notation must be read with this qualification.

A vector field along a smooth map between manifolds $f:M\to N$ assigns $Y_p\in T_{f(p)}N$ smoothly to each $p$. It is a section of the pullback tangent bundle, not necessarily a vector field on $N$. Along a curve $\gamma$, the velocity $\dot{\gamma }$ is such a section.