Geometry-kinematics duality

We propose a mapping between geometry and kinematics that implies the classical equivalence of any theory massless bosons -- including spin exhibiting arbitrary derivative or potential interactions to nonlinear sigma model (NLSM) with momentum-dependent metric in field space. From this kinematic we construct corresponding connection, covariant derivative, curvature, all which transform appropriately under general redefinitions, even derivatives. show explicitly how tree-level on-shell scattering amplitudes are equal those NLSM via replacement kinematics. Lastly, describe recently introduced geometric soft theorem NLSM, universally encodes leading subleading scalar theorems, also captures photon theorems.