Realization of doubly special relativistic symmetries in Finsler geometries

Finsler geometry is a well-known generalization of Riemannian geometry which allows us to account for a possibly nontrivial structure of the space of configurations of relativistic particles. Here we establish a link between Finsler geometry and the sorts of models with curved momentum space and doubly special relativistic relativistic symmetries which have been of interest recently in the quantum-gravity literature. We use as a case study the much-studied scenario which is inspired by the -Poincar\'e quantum group and show that the relevant deformation of relativistic symmetries can be implemented within a Finsler geometry.