Connection along a curve

Definition

Let $M$ be a differential manifold and $E$ be a vector bundle over $M$. Let $\gamma :[0,1]\to M$ be a smooth curve in $M$. A connection along $\gamma$, of $E$, is defined as follows: it is a map $D/dt$ from the space of sections of $E$ along $\gamma$, to itself, such that:

$DV/dt+DW/dt=D(V+W)/dt$

and for $f:[0,1]\to \mathbb {R}$ we have:

$D(fV)/dt=fDV/dt+Vdf/dt$

where $df/dt$ is usual real differentiation.

Facts

Connection gives connection along a curve

Given a connection on the whole vector bundle $E$, we can obtain a connection along the curve $\gamma$. Simply define:

$DV/dt=\nabla _{\gamma '(t)}V$

where $\gamma '(t)$ is the tangent vector to $\gamma$ at $\gamma (t)$. This can also be viewed as the pullback connection for the map $\gamma$ (which we might restrict to the open interval $(0,1)$, for convenience).

However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.