Before revision:  For local infinitesimal self-motion, the change of the vector is determined by the directional derivative of the recurrent network, and the norm of the directional derivative captures the metric of the path integrationmodel. We identify an isotropic condition on the norm of the directional derivative of the recurrent network , so that the local change of this vector is a conformal embedding of the local self-motion.
After revision:  For local infinitesimal self-motion, the change of the vector . We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding , so that the local change of this vector is a conformal embedding of the local self-motion.