# 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 $f:[0,1] \to \R$ we have:

$D(fV)/dt = fDV/dt + V df/dt$

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.