The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of "connecting" the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a "connection". In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, "vice versa", parallel transport is the local realization of a connection.