Connection along a curve
Let be a differential manifold and be a vector bundle over . Let be a smooth curve in . A connection along , of , is defined as follows: it is a map from the space of sections of along , to itself, such that:
and for we have:
where is usual real differentiation.
Connection gives connection along a curve
Given a connection on the whole vector bundle , we can obtain a connection along the curve . Simply define:
where is the tangent vector to at . This can also be viewed as the pullback connection for the map (which we might restrict to the open interval , 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.