Connection along a curve

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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.