  A Jacobi field on a Riemannian manifold M is defined along a geodesic. We
generalize this notion to an arbitrary smooth curve, and call it an
infinitesimal isometry along the curve. We give two approaches to this: 1)
compute the complete prolongation of the Killing equation and then restrict to
the curve, and 2) compute the variational equations of a rigid motion of the
curve. This results in Killing transport along the curve, which is parallel
transport for a related connection on the jet bundle J(TM). We study the
curvature and holonomy of this connection. In particular, in dimension two the
curvature is the local obstruction to infinitesimal isometries on M.
