  We define a perturbatively calculable quantity--the on-shell
correlator--which furnishes a unified description of particle dynamics in
curved spacetime. Specializing to the case of flat and anti-de Sitter space,
on-shell correlators coincide precisely with on-shell scattering amplitudes and
boundary correlators, respectively. Remarkably, we find that symmetric
manifolds admit a generalization of on-shell kinematics in which the
corresponding momenta are literally the isometry generators of the spacetime
acting on the external kinematic data. These isometric momenta are
intrinsically non-commutative but exhibit on-shell conditions that are
identical to those of flat space, thus providing a common language for
computing and representing on-shell correlators which is agnostic about the
underlying geometry. Afterwards, we compute tree-level on-shell correlators for
biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn
that color-kinematics duality is manifested at the level of fields under a
mapping of the color algebra to the algebra of gauged isometries on the
spacetime manifold. Last but not least, we present a field theoretic derivation
of the fundamental BCJ relations for on-shell correlators following from the
existence of certain conserved currents in BAS theory and the NLSM.
